A 


PRACTICAL TREATISE 

ON 

ARITHMETIC, 

WHEREIN 

EVERY PRINCIPLE TAUGHT IS EXPLAINED IN A SIMPLE AND 

OBVIOUS MANNER j 


CONTAINING 


NUMEROUS QUESTIONS, 

A*’D 

COMBINING THE USEFUL PROPERTIES OF FORMER WORKS, 
WITH THE MODERN IMPROVEMENTS. 

BEING 

A COMPLETE SYSTEM. 

TO WHICH IS ADDED 

TWO METHODS OF BOOK-KEEPING, 

WIT.I EXAMPLES FOR EXERCISE. 


Bt GEORGE LEONARD, Jr. 


SECOND EDITION, STEREOTYPED. 


t *2 . ** 

BOSTON; 

OTIS, BROADERS, AND COMPANY. 

NEW YORK, ROBINSON, PRATT, & CO , AND COLLINS, KEESE, & CO.; 
PHILADELPHIA, THOMAS, COWPERTIIWAIT, & CO.; BALTIMORE, 
CUSHING & BROTHER; CINCINNATI, E. LUCAS & CO.; LOUISVILLE, 
MORTON <fc GRISWOLD. 


1841 . 









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A 


PRACTICAL TREATISE 

ON 

ARITHMETIC, 


WHEREIN 

EVERY PRINCIPLE TAUGHT IS EXPLAINED IN A SIMPLE AND 

OBVIOUS MANNER 5 

CONTAINING 

NUMEROUS QUESTIONS, 

AND 

COMBINING THE USEFUL PROPERTIES OF FORMER WORKS, 
WITH THE MODERN IMPROVEMENTS. 

X. 

BEING 

A COMPLETE SYSTEM. 

TO WHICH 13 ADDED 

TWO METHODS OF BOOK-KEEPING, 

WITH EXAMPLES FOR EXERCISE. 


By GEORGE LEONARD, Jr. 


SIXTH EDltlON, STEREOTYPED. 

. * > 

' , 


: ", •' ' 

l * ' 

1 <• n » 

BOSTON; 

OTIS, BROADERS, AND COMPANY. 

NEW YORK, ROBINSON, PRATT, & CO, AND COLLINS, KEESE, «fc CO.; 
PHILADELPHIA, THOMAS, COWPERTHWAIT, & CO.; BALTIMORE, 
CUSHING <fe BROTHER; CINCINNATI, E. LUCAS & CO.; LOUISVILLE, 
MORTON & GRISWOLD. 


1841 






4-af\ 





Entered according to Act of Congress, in the year 1840, by 
George Leonard, Jr., 

in the Clerk’s office of the District Court for the District of Massachusetts. 






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'■'S'f'- TE OF 

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CAMBRIDGE j 

STEREOTYPED BY 

FOLSOM, WELLS, AND THURSTON. 







PREFACE 


The manner of teaching arithmetic was formerly very 
different from that employed at the present time. Certain 
arbitrary precepts or rules were stated, according to 
which the scholar performed the examples, remaining in 
entire ignorance of the propriety of his operations. Such 
rules are soon forgotten ; no person regards them, but 
solves the questions that occur in business, by means of 
principles suggested by common sense. There seems to 
be an obvious improvement, then, in late works, where 
the scholar, in learning the science, is taught to investi¬ 
gate and apply those principles on which he must depend 
in practice. 

This treatise combines the conciseness of the old sys¬ 
tem with the advantages of the new. It commences in a 
very simple manner, so as to be readily understood by a 
person of moderate capacity, having no previous knowl¬ 
edge of the subject. As it advances, the examples and 
questions are so arranged, that the scholar is led by im¬ 
perceptible degrees to discover new principles. The rea¬ 
sons for every rule and operation are made obvious, and 
when exolanations are necessary, great care has been 
taken to render them very lucid and concise. 

The subjects are arranged and discussed in a more nat¬ 
ural order than that usually adopted ; for instance, even in 
the late improved arithmetics, Fractions are partially de¬ 
scribed in Division ; Federal Money follows immediately 
after Division, so that many of the principles of Decimal 
Fractions are employed before they can be well explained 



4 


PREFACE. 


or understood ; Compound Numbers, being usually put 
before Fractions, cannot be described in one place, and 
are resumed again in the latter part of Fractions, under 
heads called Reduction of Fractions and Reduction of 
Decimals. When subjects are divided in this way, and 
dissimilar ones jumbled together, the learner is greatly 
confused and retarded. On the contrary, in this treatise, 
Federal Money and Compound Numbers succeed Com¬ 
mon and Decimal Fractions ; whence the scholar, in 
Federal Money, learns no new rule, but merely applies 
the principles that govern Decimal Fractions ; and in 
Compound Numbers, he employs Common and Decimal 
Fractions in the same manner as in other cases. 

There are many similar improvements ; no subject 
being introduced until every thing necessary to 6e known 
before, has been explained in its proper place. No article 
is mutilated or superficially described; but every thing im¬ 
portant to be known concerning it is fully and fairly stated. 

As examples to be performed in the mind furnish a 
very useful and necessary exercise, we have given an 
adequate number, and have interspersed them throughout 
the work, in appropriate places, so as to afford a pleasing 
variety, and illustrate the different parts. 

At the bottom of each page are copious questions, 
having a phraseology similar to that of their respective 
answers, so that the learner sees at once what should be 
committed to memory. The great number of questions, 
their arrangement at the bottom of each page, and their pe¬ 
culiar adaptation to the required answers, save much labor 
and vexation to the instructer, as well as to the scholar; 
and in connexion with the simplicity of the work, and the 
regular gradation by which it proceeds from the obvious 
to the more abstruse, render it very convenient for lha 
purposes of self-instruction. 


PREFACE. 


5 


This Arithmetic is well calculated for the wants of the 
farmer and mechanic, being short, plain, and practical. 
The merchant will find no work that can be studied with 
greater advantage, or which contains more that is really 
useful for his purpose, while the mode of reasoning and 
the general plan are well suited to the scientific or literary 
student. 

The articles on Square Root, Cube Root, Mensura¬ 
tion, and Simple Machines, are explained, it is hoped, 
with much greater clearness and precision than in any 
similar work. Book-Keeping, and the Forms of Notes, 
Bonds, Orders, Receipts, &c., are treated in a manner 
quite new and original ; for there is not only a clear and 
accurate account of their use, with the necessary exam¬ 
ples, but the scholar is required to write, in a proper 
form, many of the transactions usual in business. 

This work iS intended to be a complete treatise on 
arithmetic. It contains every useful rule that can be in¬ 
troduced with propriety, and commences in a manner so 
simple as to render the study of an introduction unneces¬ 
sary. However, an introduction for small scholars may 
be useful, to familiarize their minds with the subject, and 
preserve a more valuable book from being torn and de¬ 
faced. 

The person who desires a competent knowledge of 
arithmetic should study every part of this work, the com¬ 
mencement, as well as the rules of more direct applica¬ 
tion ; however, if the scholar intends to become a farmer 
or mechanic, and has but a short time to devote to the 
subject, he can omit all after Fellowship, except Book- 
Keeping, as of secondary importance. Still, the articles 
on Mensuration and Simple Machines, as well as many 
others, are very useful in practice, and furnish an excellent 
discipline for the mind. Even those on Money, and 


6 


PREFACE. 


Weights and Measures, though intended chiefly for the 
merchant, should be read by all who desire to understand 
books of travels, histories, or even a common newspaper. 

Many examples marked in this work for the slate, are 
readily performed in the mind, and the scholar should be 
required to solve as great a number of these mentally as 
may be deemed expedient. It is important that many 
examples should be performed both on the slate and in 
the mind, since one operation proves the correctness of 
the other. Men of business often test a calculation in this 
manner. 

We cannot forbear mentioning here an easy way to 
examine the work performed on the slate. Let each 
scholar preserve his figures, and when the class is 
called out to recite, the first should be required to de¬ 
scribe the manner in which he obtained the answer to a 
certain example in the lesson. Those \frho have found 
the answer in the same way should then hold up their 
slates, after which the teacher pronounces the w r ork right 
or wrong. He now directs those who have taken a dif¬ 
ferent course, to explain it, and bestows praise or blame 
on it ; as he thinks proper. The next scholar is then 
questioned concerning another example, and so on through 
the lesson. 

This method, if adopted, will oblige the learner ac¬ 
tually to work out each result which he brings forward, 
and will likewise prevent his guessing at a method of 
solving any question, since he must presently give the 
reasons for each operation. 


CONTENTS. 


Page 

Numeration,. 9 

Addition,. 20 

Subtraction,. 28 

Multiplication,... 35 

Division,. 45 

Promiscuous Questions in Addition, Subtraction, Mul¬ 
tiplication and Division,. 56 

Common Fractions, .... . 59 

Decimal Fractions,.... 81 

Promiscuous Questions in Fractions,. 93 

Ratio,. 94 

Federal Money,. 97 

Compound Numbers ; Tables,. 104 

Reduction of Compound Numbers,. 110 

Addition of Compound Numbers,. 122 

Subtraction of Compound Numbers,. 125 

Multiplication of Compound Numbers,. 127 

Division of Compound Numbers,. 128 

Multiplication and Division of Compound Numbers 

by Fractions and Mixed Numbers,. 130 

Promiscuous Questions in Federal Money and Com¬ 
pound Numbers,. 131 

Percentage,. 134 

Commission,. 136 

Stocks,. 137 

Bankruptcy,. 138 

Loss and Gain,... 139 

Draft and Tare,. 140 

Duties,. 144 

Simple Interest,. 145 

Compound Interest,. 158 

Discount,. 161 

Banking,. 163 

Equation of Payments,. 164 

Promiscuous Questions in Percentage, Commission, 

Stocks, and the similar rules,. 167 

Rule of Three,. 170 




































8 CONTENTS. 

Page 

Rule of Three Compound,. 178 

Chain Rule,. 180 

Barter,.*. 181 

Assessment of Taxes,. 182 

Simple Fellowship,. 185 

Compound Fellowship,. 187 

Insurance,. 189 

General Average,. 192 

Alligation Medial, .... 196 

Alligation Alternate,..*. 197 

Promiscuous Questions in Rule of Three, Fellowship, 

Insurance, &c.,. 203 

Mensuration,. 206 

Gauging,. 221 

Tonnage of Vessels,.223 

Square Root,.224 

Cube Root,. 233 

Specific Gravity,. 241 

The Lever,.246 

The Wheel and Axle,. 250 

The Pulley,..252 

The Inclined Plane,.254 

The Screw,. 255 

The Wedge,. 257 

Promiscuous Questions in Mensuration, Square Root, 

Cube Root, &c., .. 259 

Progression by Difference,. 261 

Progression by Quotient,. 267 

Progression by Quotient applied to Compound Interest, 272 

Annuities at Simple Interest,. 274 

Annuities at Compound Interest,.,. 276 

Promiscuous Questions in Progression by Differ¬ 
ence, &c.,. 279 

Money,. 280 

Coins, ... 287 

Exchange,. 291 

Weights and Measures,. 297 

Book-Keeping,.. 305 

Business Forms,. 320 

Signs, . 333 

Repeating Decimals,.. 

Duodecimals,.. 

Proportion,. 338 









































NUMERATION 

Lesson 1. 


1 one 

2 two 

3 three 

4 four 

5 fire 

6 six 

7 seven 

8 eight 

9 nine 

10 ten 

11 eleven 

12 twelve 

13 thirteen 

14 fourteen 

15 fifteen 

16 sixteen 

17 seventeen 

18 eighteen 

19 nineteen 

20 twenty 

21 twenty-one 

22 twenty-two 

23 twenty-three 

24 twenty-four 

25 twenty-five 

26 twenty-six 

27 twenty-seven 

28 twenty-eight 

29 twenty-nine 

30 thirty 

31 thirty-one 

32 thirty-two 

33 thirty-three 

34 thirty-four 


Count one hundred. 

35 thirty-five 

36 thirty-six 

37 thirty-seven 

38 thirty-eight 

39 thirty-nine 

40 forty 

41 forty-one 

42 forty-two 

43 forty-three 

44 forty-four 

45 forty-five 

46 forty-six 

47 forty-seven 

48 forty-eight 

49 forty-nine 

50 fifty 

51 fifty-one 

52 fifty-two 

53 fifty-three 

54 fifty-four 

55 fifty-five 

56 fifty-six 

57 fifty-seven 

58 fifty-eight 

59 fifty-nine 

60 sixty 

61 sixty-one 

62 sixty-two 

63 sixty-three 

64 sixty-four 

65 sixty-five 

66 sixty-six 

67 sixty-seven 


68 sixty-eight 

69 sixty-nine 

70 seventy 

71 seventy-one 

72 seventy-two 

73 seventy-three 

74 seventy-four 

75 seventy-five 

76 seventy-six 

77 seventy-seven 

78 seventy-eight 

79 seventy-nine 

80 eighty 

81 eighty-one 

82 eighty-two 

83 eighty-three 

84 eighty-four 

85 eighty-five 

86 eighty-six 

87 eighty-seven 

88 eighty-eight 

89 eighty-nine 

90 ninety 

91 ninety-one 

92 ninety-two 

93 ninety-three 

94 ninety-four 

95 ninety-five 

96 ninety-six 

97 ninety-seven 

98 ninety-eight 

99 ninety-nine 
100 one hundred 


10 


NUMERATION. 


How many ones or units make 10? 10 and how many 
ones or units make 12 ? 10 and how many units make 13 ? 
14 ? 16 ? 19 ? 15 ? 18 ? 17 ? 11 ? 

How many tens are there in 20 ? In 30 ? 40 ? 60 ? 80 ? 
70 ? 50 ? 90 ?' 100 ? 

How many tens and units are there in 21 ? In 23 ? 28 ? 

26 ? 32 ? 35 ? 37 ? 44 ? 49 ? 41 ? 53 ? 57 ? 62 ? 65 ? 

68 ? 71 ? 76 ? 79 ? 85 ? 87 ? 88 ? 92 ? 94 ? 99 ? 

Observe that thirteen is a contraction of three and ten ; 

fourteen, of four and ten ; fifteen, of five and ten ; sixteen, 
of six and ten ; seventeen, of seven and ten ; eighteen, of 
eight and ten ; and nineteen, of nine and ten. 

Twenty is a contraction of two tens ; thirty, of three 
tens ; forty, of four tens ; fifty, of five tens ; sixty, of six 
tens ; seventy, of seven tens ; eighty, of eight tens ; and 
ninety, of nine tens. 

What do you call 10 and 1 ? 10 and 3 ? 10 and 7 ? 10 
and 9 ? 2 tens ? 2 tens and 1 ? 2 tens and 5 ? 2 tens and 
7 ? 3 tens ? 3 tens and 2 ? 3 tens and 8 ? 4 tens ? 4 tens 
and 6 ? 5 tens ? 5 tens and 3 ? 5 tens and 5 ? 6 tens ? 6 
tens and 4 ? 7 tens ? 8 tens ? 8 tens and 6 ? 9 tens ? 9 
tens and 2 ? 9 tens and 9 ? 10 tens ? 

How do you write one, two, three, &c., up to ten, in 
figures, on your slate ? How do you write in figures, on 
your slate, fourteen, sixteen, seventeen, twenty, twenty- 
seven, thirty, thirty-three, thirty-six, forty, forty-one, forty- 
five, fifty, fifty-two, fifty-four, sixty, sixty-nine, seventy, 
seventy-one, seventy-eight, eighty, eighty-three, eighty- 
four, ninety, ninety-six, one hundred ? 

Lesson 2. 

If we have a great many things to number, say a large 
quantity of silver dollars, we count out a heap containing 
one hundred in the preceding manner ; we then count out 
another heap of one hundred, making with the first, two 
hundred, and so proceed counting out heaps of one hun¬ 
dred dollars each, making three hundred, four hundred, 


What is thirteen a contraction of? Fourteen ? Seventeen ? Fifteen? 
Nineteen ? Sixteen ? Eighteen ? 

What is twenty a contraction of ? Thirty? Sixty? Eighty? Forty? 
Seventy ? Fifty ? Ninety ? 



NUMERATION. 


11 


five hundred, six hundred, seven hundred, eight hundred, 
nine hundred, till we get ten hundred, which is called a 
thousand. 

Ten heaps of a thousand dollars each make ten thou¬ 
sand ; ten heaps of ten thousand each make one hundred 
thousand ; and ten heaps of one hundred thousand each 
make ten hundred thousand, or a thousand thousand, called 
a million. 

Ten heaps of a million each make ten millions ; ten 
heaps of ten millions each make one hundred millions ; 
and ten heaps of one hundred millions each make one 
thousand millions, called a billion. 

In like manner, a thousand billions make a trillion ; a 
thousand trillions a quatrillion, and so we go on to quin- 
tillions, sextillions, septillions, octillions, nonillions, de- 
cillions, &c., each number being a thousand times the pre¬ 
ceding one. 

How many dollars are there in 2 heaps of a thousand 
dollars each ? 3 heaps ? 4 heaps ? 5 heaps ? 6 heaps ? 7 
heaps ? 8 heaps ? 9 heaps ? 10 heaps ? 

How many dollars are there in 2 heaps of ten thousand 
dollars each ? 3 heaps ? 4 heaps ? 5 heaps ? 6 heaps ? 7 
heaps ? 8 heaps ? 9 heaps ? 10 heaps ? 

How many dollars are there in 2 heaps of one hundred 
thousand dollars each ? 3 heaps ? 4 heaps ? 5 heaps ? 6 
heaps ? 7 heaps ? 8 heaps ? 9 heaps ? 10 heaps ? 

Make one thousand marks on your slate. Show how 
many make twenty-five, fifty, seventy-five, ninety-seven, 
one hundred and seventeen, one hundred and sixty, one 
hundred and eighty-four, two hundred, two hundred and 
seven, three hundred, four hundred, five hundred, five hun¬ 
dred and fifty, six hundred, seven hundred, seven hundred 
and seventy, eight hundred, eight hundred and fifty, nine 
hundred. • 


How should we proceed to count a thousand silver dollars ? 

How many do ten heaps of one thousand dollars each make ? Of 
ten thousand dollars each ? Of one hundred thousand dollars each ? 
Of a million each P Of ten millions each ? Of one hundred millions 
each ? 

How many billions make a trillion ? How many trillions a quatril¬ 
lion ? How then do we go on ? How large is each number ? 



NUMERATION. 


12 

One hundred and one is 
made in figures thus . 101 
One hund. and two . . . 102 
One hund. and three . . 103 
One hund. and four ... 104 
One hund. and five ... 105 
One hund. and six .... 106 
One hund. and seven . . 107 
One hund. and eight . . 108 
One hund. and nine ... 109 
One hund. and ten .... 110 
One hund. and eleven .111 
One hund. and twelve .112 
One hund. and thirteen 113 
One hund. and fourteen 114 
One hund. and twenty- 

three ..:'. . . 123 

One hund. and thirty- 

seven .137 

One hund. and forty. . . 140 


One hund. and fifty-six. 156 
One hund. and eighty- 

one .181 

Two hundred. 200 

Two hund. and one ... 201 

Two hund. and two . . . 202 

Two hund. and thirty- 

one .231 

Three hundred.300 

Three hund. and seven¬ 
ty-six .376 

Four hundred.400 

Five hundred.500 

Six hundred.600 

i Seven hundred.700 

Eight hundred.800 

Nine hundred.900 

One thousand. 1,000 

Two thousand five hun¬ 
dred . 2,500 


Lesson 3.' 

The preceding numbers, and all others, are expressed by 
only ten figures figures 1, 2, 3, 4, 5, 6 , 7, 8 , 9, and 0. 
0 always stands for nothing, and is called nought. 

If you have some cents, and write a figure 5 to express 
the number, how many will you appear by this to have ? 

Does a single figure then, say 5, stand for five units, five 
tens, five hundreds, or what ? 

When you write any single figure, say 1 , what do you 
mean, one unit, one ten, one hundred, or what ? What do 
you mean when you write 2 ? 3 ? 4 ? 5 ? 6 ? 7 ? 8 ? 
9 ? 0 ? 

Write fourteen on your slate in figures. What does the 
4 at the right hand side stand for ? What does the 1 stand 
for, one unit, one ten, one hundred, or what ? 


HoV do you make one hundred and one in figures, on your slate ? 
One hundred and two ? So proceed with questions through the other 
numbers, and embrace some of the intermediate numbers not here ex¬ 
pressed. 

By how many figures are the preceding numbers and all others ex¬ 
pressed ? What are these figures ? What does 0 always stand for ? 
What is it called ? 



















NUMERATION. 13 

Write ten on your slate in figures. Does the 0 stand 
for any number ? What does the 1 stand for ? 

Write thirty-seven on your slate in figures. What does 
the 7 stand for ? What does the 3 stand for ? 

Suppose that instead of writing 37 you write 73, what 
number do you express ? What does the 3 now stand for ? 
The 7 ? 

Write one hundred and thirty-eight on your slate in 
figures. What does the 8 stand for ? The 3 ? The 1 ? 
On which side of 138 do you find the 8 units, on the right 
hand side, or on the left hand side ? How many figures left 
o.f the units do you find the 3 tens ? The 1 hundred ? 

Write one hundred on your slate in figures. There be¬ 
ing no odd ones or units, what stands in the units’ place ? 
There being no odd tens, what stands in the tens’ place ? 
What does the 1 stand for ? 

Write six hundred and four on your slate in figures. 
What does the 4 stand for ? What shows there are no 
tens ? What does the 6 stand'for ? 

Write one thousand on your slate in figures. There 
being no odd units, tens, or hundreds, what stands in the 
units’ place, the tens’ place, and the hundreds’ place ? 
What does the 1 stand for ? How many figures left of the 
units is the 1 in the thousands’ place ? 

When one figure stands by itself, as 6 , what does it 
mean, 6 units, 6 tens, 6 hundreds, or what ? 

A figure standing at the right of another, or of others, as 
8 in 958, is units, tens, hundreds, or what ? A figure stand¬ 
ing one place left of units, as 5 in 958, is units, tens, hun¬ 
dreds, or what ? A figure standing two places left of units, 
as 9 in 958, is units, tens, hundreds, or what ? A figure 
standing three places left of units, as 2 in 2,500, is units, 
tens, hundreds, or what ? 

So we find the units at the right hand side of a number , the 
tens one place left of units , the hundreds two places left of units, 
and the thousands three places left of units. 

Lesson 4. 

How many times is 1 in 10 ? How many times is 10 in 
100 ? How many times is 100 in 1,000 ? How many times 

In what part of a number do we find the units ? The tens ? The 
hundreds ? The thousands ? 

2 



14 


NUMERATION. 


is 2 in 20 ? How many times is 3 in 30 ? 4 in 40 ? 5 in 
50 ? 6 in 60 ? 7 in 70 ? 8 in 80 ? 9 in 90 ? 

How many times is 20 in 200 ? How many times is 30 in 
300 ? 40 in 400 ? 50 in 500 ? 60 in 600 ? 70 in 700 ? 80 
in 800 ? 90 in 900 ? 

In 11 , how much more does the 1 in the tens’ place stand 
for than the 1 in the units’ place ? 

In 22 , how much more does the 2 in the tens’ place stand 
for than the 2 in the units’ place ? 

In 55, how much more does the 5 in the tens’ place stand 
for than the 5 in the units’ place ? 

In 99, how much more does the 9 in the tens’ place stand 
for than the 9 in the units’ place ? 

In 330, how much more does the 3 in the hundreds’ place 
stand for than the 3 in the tens’ place ? 

In 880, how much more does the 8 in the hundreds’ place 
stand for than the 8 in the tens’ place ? 

It is just so in all cases ; therefore 

J1 figure at the left of another 4 stands for ten times as much 
as it would in the place of that other figure. 

From the preceding principles we see that 

7 stands for seven. 

27 stands for twenty-seven. 

127 stands fpr one hundred and twenty-seven. 

5,127 stands for five thousand, one hundred and twenty- 
seven. 

35,127 stands for thirty-five thousand, one hundred and 
twenty-seven. 

835,127 stands for eight hundred and thirty-five thou¬ 
sand, one hundred and twenty-seven. 

4,835,127 stands for four millions, eight hundred and thirty- 
five thousand, one hundred and twenty-seven. 


A figure at the left of another stands for how much more than it 
would in the place of that other figure ? 

How do you make twenty-seven, in figures, on your slate ? One 
hundred and twenty-seven ? Five thousand, one hundred and twenty- 
seven ? Thirty-five thousand, one hundred and twenty-seven ? Eight 
hundred and thirty-five thousand, one hundred and twenty-seven ? 
Four millions, eight hundred and thirty-five thousand, one hundred and 
twenty-seven ? 



NUMERATION. 


1 5 


NUMERATION - TABLE. 


The following Numeration Table shows plainly the man¬ 
ner in which figures increase in value as they are placed 
further and further to the left. 


3 cs . 
o „ a. 

g .5 » 
® bn-a 

to 4. ci 
4; .o a> 
C3 

K • •' 

©.Sll 

«» be” 

Og| 

to 

<u ^ £ 

s>° 

cd Yl 
jn3 w 

^ a> * 
0,0 


•s 

a 


>-> >3 

g£ 

* s 

0) -j? 

-a e 

H.« 

43 


S3 

O 

J 

h4 

5 

<1 

6 

cf 

6 , 


Clj 

S3 

cr 

* 

a3 


cn 

S3 

# o 

r33 co 
P c3 

~ o 
P-. ^3 
O 35 
® £ 
a» ^ 

P O 

“H tn 
g S3 


ffiHH 
4 8 5, 


-3 

.fl 

cd 


is* ^3 


ai 
S3 

g 

P H '-I 

H3 JS r3 
S3 bJ3i3 


S3 

Pi 

S3 

,o 


CO 

S3 

O 

SS3 tn 
42 S3 

P, .2 

O =3 

co 13 
'-O tn 
0 ) Pi £ 
Pi O O 

^ S d 

WHpq 


CO 

S3 

o 

rss co 

s g 

P, S=3 

41« 

poo 
"2 ® d 

J g d 

Bps 


CO 

r O 


a ® 

3 

I § 

£ 3 

» 

co Fh Q 

CJ Pi 
P O 

Is 2 w 

■5 a> K 

Bpp 


co 

D 

O 


co 

T3 

03 

P 

H3 

S3 

3 

ffi 


to 

H 

s 

£> 

p 

O 

ai cfl 
£3 W 
03 23 

HO 


0 2 7, 9 1 3, 5 4 0, 2 7 6 


03 

> 

<S3 


co 

S3 

*2 
y, -g 3 

§ 03 

£ O 

03 

‘ > 

03 

CO 


n3 
S3 

3 

nS 

03 

'O .£3 

15 S 

43 S3 

03 

03 03 

S3 ^ 


CO 

S3 
O 

p rs3 


nS «rT 
2 

-§ t; g 


nS 

i 

1 i* 


Numbers “to be easily read, should be divided like these, 
by commas, into parts of three figures each, beginning at 
units. 

The ten figures, 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0, together 
with the present manner of using them, were invented in 
India, many years ago. Some knowledge of them was 
given us by the Arabs, about 800 years since, they having 
obtained their information from the Indians. All civilized 
people now reckon with these figures. 


Copy the Numeration Table on your slate, and name the places oc¬ 
cupied by each figure^ beginning at ones or units. Read the figures in 
the table you have copied, beginning at six quatrillions. 

How should numbers be divided to be easily read ? 

Where were the ten figures, 1.2, 3, 4, 5, fi, 7, 8, 9, and 0 invented ? 
Who gave us some knowledge of them ? How long since ? From whom 
did they obtain their information? What people now reckon with these 
figures ? 





16 


NUMERATION. 


Lesson 5. 


Copy the following numbers on your slate in words. 


2 .708 

16 . 6,348,745,877 

17 . 45,001 

3 . 2,010 

18. 1,112 

6 . 1 , 000,000 

7. 9,045,275 

10 .370,004 

19 .200,004 

20 .75,000,473,000 

21 .800,111,000,333 

22 .3,756,235,974,212 

23 . 111 , 111,222 

24 .505,505,505,555 

25 .30,303 

14 . 77,000,691 

26. 10,001 

27 .476,766 

28 . 45,000,000 

29 .347 


30. 901 


When you write a number in figures that has no odd 
units, as one hundred and fifty, what do you put in the 
units’ place ? When there are no odd tens, as in one hun¬ 
dred and five, what do you put in the tens’ place. When 
there are no odd hundreds, as in four thousand and forty, 
what do you put in the hundreds’ place ? When there are 
no odd thousands, as in forty thousand, five hundred and~ 
twelve, what do you put in the thousands’ place ? 

If 0 stands at the left of a figure, say of figure five ; 
thus, 05, how many tens are there in these figures ? How 
many units ? Can the 0 be omitted, then, without chang¬ 
ing the value of the number ? 

If you put 0 at the right of figure five, thereby causing 
the 5 to stand in the tens’ place ; thus, 50, what do these 
figures now stand for ? How many times 5 do they stand 
for ? If you put 0 at the right of 50 ; thus, 500, what do 
these figures stand for ? How many times 50 do they ex¬ 
press ? 

Therefore , 0 placed at the left of a number, has no effect , 
and may be omitted ; placed at the right of a number, it in¬ 
creases it ten times. 


What effect has 0 placed at the left of a number ? Placed at the right 
of a number ? 


































numeration. 


nf 


Lesson 6. 

Copy the following numbers on your slate in figures, and 
divide them into parts so as to be easily read. 

1. Six hundred and five. 

2 . Nine hundred and eighty-three. 

3. Five thousand, four hundred and thirty-two. 

4. Eight thousand, and ninety-five. 

5. Forty thousand, and one. 

6 . Sixty-five thousand. 

7. Eighty-one thousand, two hundred and twelve. 

8 . One hundred and twenty-five thousand, two hundred. 

9. Four hundred and seventy-seven thousand, eight hun¬ 

dred. 

10 . One million. 

11 . One million, one thousand, five hundred. 

12 . Fourteen millions, seven hundred and five thousand, 

six hundred and forty-one. 

13. Eighty-seven millions, two hundred and seventy-five 

thousand, one hundred and twelve. 

14. Three hundred and thirty-three thousand, and three. 

15. Nine hundred millions. 

16. Two billions, seventeen millions, one thousand, and 

thirteen. 

17. Twenty-five billions. 

18. Four hundred and forty billions, three hundred and 

twenty-one thousand. 

19. Eight trillions, two hundred thousand. 

20 . Five hundred and forty-seven trillions, three hundred 

and eleven millions, eight hundred and thirty-four 
thousand, two hifndred and eighty-eight. 

21 . Two thousand, three hundred and four. 

22 . Fifty-five trillions, eight hundred and seven billions. 

23. Six hundred trillions, four hundred and fourteen bil¬ 

lions, and three. 

24. Forty-five billions, threfe hundred and twenty-seven 

millions. 

.25. Ten thousand, and ten. 

26. One hundred thousand, two hundred. 

27. Forty-four thousand, four hundred and forty-four. 

28. Five billions, and two. 

29. Twenty-seven millions. 

30. Four thousand, two hundred and twelve. 

2 # 


18 


NUMERATION. 


Lesson 7. 

There is another kind of numbers which it will be well 
to understand. The following table explains these num¬ 
bers, and their use. 

If you are making snow-balls, after you finish 
One, that will be the first you make ; after you finish 

Two, the last will be.the second 

Three.the third 

Four.the fourth 

Five.the fifth 

Six.the sixth 

Seven.the seventh 

Eight...the eighth 

Nine.the ninth 

Ten.the tenth 

Eleven.the eleventh 

Twelve ...the twelfth 

Thirteen... ....... .the thirteenth 

Fourteen...the fourteenth 

Fifteen..the fifteenth 

Sixteen.the sixteenth 

Seventeen.....the seventeenth 

Eighteen.the eighteenth 

Nineteen...the nineteenth 

Twenty.the twentieth 

Twenty-one.the twenty-first 

Twenty-two.the twenty-second 

Twenty-three. .the twenty-third 

Thirty.the thirtieth 

Forty..... . v .the fortieth 

Fifty.the fiftieth 

Sixty .....the sixtieth 

Seventy.the seventieth 

Eighty.the eightieth 

Ninety.the ninetieth 

One hundred...the one hundredth 

One hundred and one.the one hundred and first 


If you are making snow-balls, after you finish one, will that be the 
first, second, or third, that you make ? After you finish two, what will 
the last be? Three? Four? So proceed with questions through the 
other numbers. 


































NUMERATION. 


19 


One hundred and two. 

One hundred and three.... 

Two hundred.;. 

Three hundred. 

One thousand. 

Two thousand. 

Ten thousand. 

Ten thousand five hundred 
and twenty-six 
One hundred thousand.... 
One million. 


the one hundred and second 
. .the one hundred and third 

.the two hundredth 

.the three hundredth 

.the one thousandth 

.the two thousandth 

.the ten thousandth 

S the ten thousand five hun¬ 
dred and twenty-sixth 
the one hundred thousandth 
.the one millionth 


These numbers are called ordinal numbers; they are 
often employed in books and conversation, though not di¬ 
rectly in any calculation. 

If you are in a class, standing on the floor, and the mas¬ 
ter counts the scholars in the class, beginning at the head, 
and you are counted nine, what will be your place in the 
class ; the first, second, or what ? What will your place in 
the class be if you are counted 5 ? 6? 10? 13? 

If the page in the book where you are reading, is mark¬ 
ed 31^ will that page be the first, second, or third, in the 
book ? What page will it be ? If the page is marked 
52, what page of the book will it be ? What if it is mark¬ 
ed 45 ? 28 ? 14 ? 12 ? 8 ? 2 ? 60 ? 78 ? 91 ? 101 ? 300 ? 
500 ? 756 ? 1,000 ? 

A man counted out 10,000 dollars ; what do you call 
the last dollar ; the one hundredth, the one thousandth. or 
what ? What do you call the last dollar after he has 
counted out 12,000 dollars ? 13,571 ? 100,212 ? 625,471 ? 
8,000,000 ? 


What are these numbers called, and how employed ? 












20 


ADDITION. 


ADDITION. 


Lesson 8. 


Tq be performed in the mind . 


Note. The learner may be taught to reckon with his fingers, or 
what is better, with twenty marks, arranged by five and tens on his 
slate; thus. 


Mill Mill HIM MM I 


1. If you have 2 quills in one hand, and 3 in the other, 
how many have you in both hands ? How many then are 2 
and 3 ? How many are 3 and 2 ? 

2. James has 4 cents in a box ; if he puts 4 cents more 
into the box, what number will it then contain ? 

3. A laborer worked for me 5 hours one day, and 3 
hours the next ; how long did he work for me in both 
days ? 5 and 3 are how many then ? 3 and 5 ? 

4. 4 men engaged on a piece of work, were joined by 6 
more ; how many were engaged on it then ? 

5. I have 5 cents, and my brother has 3 cents ; how 
many have we both ? What number of cents have we both, 
if I have 8 cents, and my brother 5 cents ? 

6. There are 7 sheep in one part of a pasture, and 7 in 
another part ; how many would there be if they were all 
together ? What number then do 7 and 7 make ? 

7. A little girl bought a picture book for 8 cents, some 
paper for 3 cents, and a quill for 2 cents ; how much must 
she pay for the whole ? 

8. William had 9 apples, when he found 3 more ; how 
many did he then have ? 

9. If you buy a barrel of flour for 6 dollars, and a 



tor 9 dollars, how many dollars must you pay for 


10. A farmer, after selling 10 pounds of butter to a man, 
had 5 pounds left ; what quantity had he at first ? What 
is the sum of 10 and 5 then ? Of 5 and 10 ? 


ADDITION. 


21 


Lessons 9 and 10. 

ADDITION TABLE. 

Note. Questions in this table should not be asked in rotation, be¬ 
cause when they are so asked the learner can answer by merely count¬ 
ing, without the least exertion of memory. 

In one part of the table we find 2 and 4 are 6, and in another part, 4 
and 2 are 6. Each operation in the table is repeated, in this way, ex¬ 
cept when 1 is added to a number, or a number i«t added to itself. Make 
the learner observe this. 


2 

and 

1 

are 

3 

5 

and 

1 

are 

6 

8 

and 

1 

are 

9 

2 

and 

2 

are 

4 

5 

and 

2 

are 

7 

8 

and 

2 

are 

10 

2 

and 

3 

are 

5 

5 

and 

3 

are 

8 

8 

and 

3 

are 

11 

2 

and 

4 

are 

6 

5 

and 

4 

are 

9 

8 

and 

4 

are 

12 

2 

and 

5 

are 

7 

5 

and 

5 

are 

10 

8 

and 

5 

are 

13 

2 

and 

6 

are 

8 

5 

and 

6 

are 

11 

8 

and 

6 

are 

14 

2 

and 

7 

are 

9 

5 

and 

7 

are 

12 

8 

and 

7 

are 

15 

2 

and 

8 

are 

10 

5 

and 

8 

are 

13 

8 

and 

8 

are 

16 

2 

and 

9 

are 

11 

5 

and 

9 

are 

14 

8 

and 

9 

are 

17 

2 

and 

10 

are 

12 

5 

and 

10 

are 

15 

8 

and 

10 

are 

18 

3 

and 

1 

are 

4 

6 

and 

1 

are 

7 

9 

and 

1 

are 

10 

3 

and 

2 

are 

5 

6 

and 

2 

are 

8 

9 

and 

2 

are 

11 

3 

and 

3 

are 

6 

6 

and 

3 

are 

9 

9 

and 

3 

are 

12 

3 

and 

4 

are 

7 

6 

and 

4 

are 

10 

9 

and 

4 

are 

13 

3 

and 

5 

are 

8 

6 

and 

5 

are 

11 

9 

and 

5 

are 

14 

3 

and 

6 

are 

9 

6 

and 

6 

are 

12 

9 

and 

6 

are 

15 

3 

and 

7 

are 

10 

6 

and 

7 

are 

13 

9 

and 

7 

are 

16 

3 

and 

8 

are 

11 

6 

and 

8 

are 

14 

9 

and 

8 

are 

17 

3 

and 

9 

are 

12 

6 

and 

9 

are 

15 

9 

and 

9 

are 

18 

3 

and 

10 

are 

13 

6 

and 

10 

are 

16 

9 

and 

10 

are 

19 

4 

and 

1 

are 

5 

7 

and 

1 

are 

8 

10 

and 

1 

are 

11 

4 

and 

2 

are 

6 

7 

and 

2 

are 

9 

10 

and 

2 

are 

12 

4 

and 

3 

are 

7 

7 

and 

3 

are 

10 

10 

and 

3 

are 

13 

4 

and 

4 

are 

8 

7 

and 

4 

are 

11 

10 

and 

4 

are 

14 

4 

and 

5 

are 

9 

7 

and 

5 

are 

12 

10 

and 

5 

are 

15 

4 

and 

6 

are 

10 

7 

and 

6 

are 

13 

10 

and 

6 

are 

16 

4 

and 

7 

are 

11 

7 

and 

7 

are 

14 

10 

and 

7 

are 

17 

4 

and 

8 

are 

12 

7 

and 

8 

are 

15 

10 

and 

8 

are 

18 

4 

and 

9 

are 

13 

7 

and 

9 

are 

16 

10 

and 

9 

are 

19 

4 

and 

10 

are 

14 

7 

and 

10 

are 

17 

10 

and 

10 

are 

20 




22 


ADDITION. 


Lesson 11. 

To be performed in the mind. 

1 . John bought an apple for 3 cents, and an orange for 4 
cents ; how many cents did he pay for both ? 

2. If you have 4 walnuts in your pocket, and 5 in your 
hat, how many have you in both places ? 

3. Henry’s father gave him 6 cents, and his sister gave 
him 6 more ? how many had he then ? 

4 . A man walked 7 miles in the forenoon, and 4 in the 
afternoon ; how far did he walk during the day ? How 
many then are 7 and 4 ? 4 and 7 ? 

5. What is the sum of 2 and 5 ? 4 and 3 ? 4 and 8 ? 5 
and 6 ? 6 and 7 ? 8 and 7 ? 8 and 9 ? 9 and 3 ? 9 and 6 ? 

9 and 9 ? 10 and 7 ? 10 and 9 ? 

6. Mark caught 4 speckled trout in a little pond, 3 in a 
brook, and a boy gave him 5 ; how many had he then ? 

7. If you have 6 marbles in your hat, 2 in one pocket, 2 
in another, and 5 in your hand, how many will you have 
if they are all put into a heap ? 

8. If you buy a picture book for 11 cents, and a top for 
4 cents, how many cents must you pay for them ? 

9. What sum do 12 and 7 make ? 15 and 3 and 2 ? 19 
and 2 ? 24 and 5 and 6 and 2 ? 37 and 6 ? 52 and 8 ? 66 
and 2 ? 70 and 9 ? 88 and 6 ? 93 and 3 and 4 ? 

10. A little boy gathered 54 cherries from one tree, 9 
from another, and he picked up 6 from the ground ; how 
many had he then ? 

Lesson 12. 

To be performed in the mind. 

1. A man had 10 dollars in his pocket, when one of his 
neighbors paid him 13 dollars ; what number of dollars 
had he then ? Explanation. How many are 10 and 10 
and 3 ? 

2. How many are 10 and 11 ? 10 and 15 > 10 and 19 ? 

10 and 27 ? 10 and 33 ? 10 and 46 ? 10 and 59 ? 10 and 
62 ? 10 and 78 ? 10 and 81 ? 10 and 95 ? 10 and 110 ? 
10 and 120 ? 10 and 359 ? 10 and 476 ? 

3. Mary bought a quire of paper for 20 cents, and a 
book for 30 cents ; how many cents must she pay for the 
paper and book ? Explanation. How many tens are 2 tens 
and 3 tens ? What are 5 tens called ? 


ADDITION. 


4. If you ride 50 miles in one day, 30 miles the next, 
and 10 miles the next, how far do you ride in the three days ? 

5. A company of 50 men were joined by 60 more ; how 
many were there then ? Explanation. What are 10 tens 
called ? What then are 10 tens and 1 ten, making 11 tens, 
called ? 

6. What is the sum of 40 and 70 ? 80 and 90 ? 90 and 
90 ? 60 and 120 ; that is, 6 tens and 12 tens ? 

7. Alexander found 17 apples under one tree, and 20 
under another; if he puts these with 5 and 2, how many 
will there be ? 

8. If a man has 30 acres of land, and buys 45 more, 
how many will he then have ? 

9. 24 bales of cotton ore piled up with 38 ; how many 
bales do both quantities make .? Explanation. 4 and 8 are 
how many ? 2 tens and 3 tens are how many ? How many 
then are 5 tens and 12 ? 

10. What is the sum of 11 and 17 ? 21 and 43 ? 52 and 
38 ? 82 and 91 ? 77 and 85 ? 40 and 18Q ? 300 and 400 ? 
9,000 and 500 ? 


Lesson 13 . 

For the Slate. 


1. A man paid 213 dollars to various creditors on Mon¬ 
day, 402 dollars on Tuesday, and 21 dollars on Wednes¬ 
day ; what was the whole sum paid ? 


Explanation. We first write 
these numbers under one another, 
with units under units, tens under 
tens, &c., and draw a line be¬ 
neath. We now add the units, 
proceeding from the bottom up ; 
thus, 1 and 2 are 3 and 3 are 6 ; 
this 6 is placed under the column 


OPERATION. 



C h & 


*** r* 

2 1 3 
402 
2 1 


6 3 6 dollars. Answer, of units. The tens are added in 
the same way, and their sum 
placed under the column of tens, and so on with the hun¬ 
dreds. 


In example 1, lesson 13, how do we write the numbers to be added ? 
What is the next thing done, before beginning to add ? What do we 
add first ? How do we proceed in adding them ? Where do we place the 
sum of the units? What is said of the tens and hundreds ? 




24 


ADDITION. 


2. If I have 100 pounds of butter, buy 201 pounds more 
of one farmer, 322 of another, and receive 1,265 in pay¬ 
ment of a debt, what number of pounds do I then have ? 

Ans. 1,888. 

3. A farmer sold 3 barrels of cider to one trader, 2 bar¬ 

rels to another, 3 barrels to a neighboring farmer, and 1 
to his blacksmith ; how many barrels did he sell to all of 
them ? Ans. 9. 

4. A young farmer began business on 23 acres of land, 
he soon after bought 12 acres, and in a year or two more 
obtained 113 by inheritance; hoW'many acres had he then ? 

Ans. 148. 

5. Four partners furnished money to purchase merchan¬ 

dise, as follows. The first 30,112 dollars, the second 23,010, 
the third 14,234, and the fourth 2,322 ; how much did they 
all furnish ? Ans. 69,678 dollars. 


Lesson 14. 


1. A rich farmer had four pieces of land ; on the first 
he had 646 sheep, on the second 300, on the third 29, and 
on the fourth 127; how many sheep had he on all the pieces? 


OPERATION. 

m 


Explanation. The column of units, 
when added, makes 22 ; that is, 2 tens 
and 2 units ; we put the 2 odd units 
in the units’ place, and carry the 2 
tens to the next column, and add 
them with the other tens. These 2 
tens and the others make 10 tens, 
that is, 1 hundred ; there being no 
odd tens, we put 0 in the tens’ place, 
and carry the 1 hundred to the next 
column, and add it with the other 
hundreds. This 1 hundred and the other hundreds, make 
11 hundred ; that is, 1 thousand and 1 hundred ; the 1 
hundred being written in the hundreds’ place, and the 1 
thousand in the thousands’ place finishes. 


■5 » « 
g S •- 

s ® a 
5S H & 
6 4 6 
3 00 
2 9 
1 2 7 


1,1 0 2 3heep. Ans. 


In example 1, lesson 14, the column of units, when added, making 
22, that is, 2 tens and 2 units, what do we put in the units’ place ? 
What is done with the 2 tens? The sum of the tens being 10; that is, 
10 tens or 1 hundred, what do we put in the tens’ place? Why ? What 
is done with the 1 hundred ? The sum of the hundreds being 11, that 
is, 1 thousand and 1 hundred, what do we put in the hundreds’ place? 
What is done with the 1 thousand ? 



ADDITION. 


25 


2. William had 8 walnuts in one pocket, 9 in another, 
and besides these he had 15 in his trunk, and 7 in the table 
drawer ; what was the whole number he had ? Ans. 39. 

3. A dealer in lumber has 37,276 feet of boards in one 

pile, 9,536 in another, 45,092 lying on his wharf, and 8,870 
in a vessel ; how many feet of boards has he in all these 
places ? " Ans. 100,774. 

4. A merchant owes one man in Boston 975 dollars, one 

in New York 483, another in New York 237, one in New 
Orleans 87, and various other persons 689 ; what is the 
amount of all his debts ? Ans. 2,471 dollars. 

5. If you buy a yoke of oxen for 75 dollars, a cart for 57, 

three cows for 88, and a plough for 10, how much must 
you pay for the whole ? Ans. 230 dollars. 

Lesson 15. 

From what precedes we get the following 

RULE FOR ADDITION. 

Write the numbers to be added under one another , with 
units under units , tens under tens , fyc., and draw a line be¬ 
neath. Add the right hand column from the bottom upwards; 
place the units in the sum of the column beneath it , and carry 
all the tens one place to the left. So proceed to add up and 
carry in all the columns. 

If you add the work twice to prove its correctness, you 
will not be apt to detect a mistake, if you proceed the 
second time the same as the first ; thus, if you are adding 
the units in example 1, lesson 14, and say, by mistake, 7 
and 9 are 15, and 6 are 21, you will be very liable to say 
7 and 9 are 15 the second time, instead of saying 7 and 9 
are 16. We can prove the work, therefore, 

By adding each column from the top dowmoards, proceed¬ 
ing in other respects as before. If the second sum be equal to 
the first, the work will generally be right . 

Note. Each example should now be proved. 


How do you write the numbers to be added ? What do you add first ? 
What do you do with the units in the sum of the column ? With the 
tens P How then do you proceed ? 

If you add the work twice to prove its correctness, when will you 
not be apt to detect a mistake? Explain this by example 1 , lesson 14 . 
How can we prove the work in Addition ? 

3 




gg ADDITION. 


Numbers to add. 

Numbers to add. 

Numbers to add. 

(1.) 

(2.) 

(3.) 

456321783 

576377 

88745 

283683001 

6892203 

343 1 

4770542 12 

7203578 

48032 

336864326 

4349 

23324 


643481 

73782 


Lesson 16. 


Numbers to add. 

Numbers to add. 

Numbers to add. 

(1) 

(2.) 

(3.) 

858 

230007321 

5 

375 

562077899 

37 

9 

125766232 

463 

4507 

440488552 

5398 

23 

234711143 

93424 

1 


643779 

456 


29944 

678 




4. A merchant owns a sloop worth 1,000 dollars, a 

schooner worth 5,675 dollars, a brig worth 8,340 dollars, 
and a ship worth 12,345 dollars; what is the value of all 
these vessels ? Ans. 27,360 dollars. 

5. A man owns one farm containing 45 acres, a second 
containing 18 acres, a third containing 156 acres, a fourth 
containing 225 acres, and a fifth containing 9 acres ; what 
number of acres are contained in the five farms ? Ans. 453. 

6. Benjamin Franklin was born in the year 1706, and 
was 84 years old when he died ; in what year did he die ? 

Ans. in 1790. 

7. A teamster hauled a quantity of wheat indifferent 

loads containing the following numbers of bushels ; 34, 
28, 27, 32, 42, 29, 33, 35, and 30 ; what was the whole 
quantily hauled ? Ans. 290 bushels. 

8. Add the following numbers, 3,575,412, 900, 8, 27, 
8,208, 450,275, 633, 44, . 65,000, and 1,225. 

Ans. 4,101,732. 

9. A merchant paid 15,255 dollars for a store, 9,237 dol¬ 

lars for a vessel, 12,676 dollars for goods, and 275 dollars 
for a horse and chaise ; how much did he pay for the 
whole ? Ans. 37,443 dollars. 

10. The owner of a coal mine agreed to furnish some 
iron manufacturers with one hundred and sixty-five thou- 








ADDITION. 


27 


sand, two hundred bushels of coal ; a merchant, with 
forty-seven thousand, six hundred and ninety-five bushels; 
another person with eight thousand, two hundred and 
seventy-nine bushels, and he wants himself three thousand, 
five hundred and forty-five bushels ; what quantity will 
supply himself and the others ? Ans. 224,719 bushels. 

Lesson 17. 

1. A farmer bought an ox wagon for 135 dollars, a yoke 
of oxen for 75 dollars, three ploughs for 27 dollars, six cows 
for 96 dollars, and thirteen sheep for 65 dollars ; what sum 
of money must he pay for the whole ? Ans. 398 dollars. 

2. If a man be 27 years old when his first son is born, 
of what age will he be when his son is 21 years old ? 

Ans. 48 years. 

3. According to the census of 1830, Maine contained 

399,462 inhabitants, New Hampshire 269,533, Vermont 
280,679, Massachusetts 610,014, Rhode Island 97,210, and 
Connecticut 297,211 ; how many inhabitants were there in 
all of these states, which are called the New England 
States ? Ans. 1,954,109. 

4. If I pay 85 dollars for a gold watch, 37 for a coat, 
6 for a hat, 5 for a pair of boots, and 275 for a horse and 
chaise, what does the whole cost me ? Ans. 408 dollars. 

5. A merchant sold a lot of coffee for 2,327 dollars, and 
lost 637 dollars on it ; how much did the coffee cost him ? 

Ans. 2,964 dollars. 

6. Charles paid twenty-five cents for some paper, fifty- 

eight cents for a penknife, twelve cents for some quills, 
and seventy-five cents for a pair of gloves ; how much did 
he pay for the whole ? Ans. 170 cents. 

As 100 cents make a dollar, how many dollars and cents 
did he pay for the whole ? 

7. What is the sum of 7, 87,455,383, 67,914,533, 

29, 456, and 500,000. Ans. 155,870,408. 

8. A merchant bought a brig for 6,236 dollars ; he paid 

614 dollars for repairs on the hull, and 869 dollars for re¬ 
pairs on the rigging ; for what price must he sell the brig 
to gain 325 dollars ? Ans. 8,044 dollars. 

9. If your debts to different persons are as follows, 2,756 
dollars, 1,000 dollars, 75 dollars, 467 dollars, 395 dollars, 
and 5,832 dollars, how much is the whole that you owe ? 

Ans. 10,525 dollars 


SUBTRACTION. 


10. If you travel 115 miles from Portland to Boston, 42 
miles from Boston to Providence, 186 miles from Provi¬ 
dence to New York, 145 miles from New York to Albany, 
16 miles from Albany to Schenectady, and 24 miles from 
Schenectady to Saratoga ; how far do you travel in going 


from Portland to Saratoga ? 


Ans. 528 miles. 


SUBTRACTION. 


Lesson 18. 


To be performed in the mind. 


1. Ip you have 5 cents, and give away 2, how many will 
rou have left ? 2 from 5 leaves how many then ? 3 from 5 
eaves how many ? 

2. Edwin took 3 walnuts from a heap that contained 6 ; 
r many did he leave ? 



3. A man walked 7 miles from home, and afterwards re¬ 
turned 4 miles; how far was he then from home ? How 
many then does 4 from 7 leave ? Why does 4 from 7 leave 
3 ? Answer. Because 4 and 3 are 7. 

4. Maria counted 8 robins on a tree, but 3 shortly flew 
away ; how many remained ? 3 from 8 leaves what num¬ 
ber then ? Why ? 

5. I had 9 cents, but soon after spent 4 of them for some 
apples ; how many did I keep ? 

6. A boy bought a pear for 3 cents, and handed the sel¬ 
ler a 10 cent piece ; how many cents must he receive 


back ? 


7. A fisherman returning home with 12 shad, sold 5 of 
them by the way ; how many were left ? How many does 
5 from 12 leave then ? 7 from 12 ? 

8. If you have 16 walnuts, and your brother has 6, which 
has the most, and how many ? 

9. A farmer who had 14 cows, sold 5 ? how many did 
he keep ? 

10. A ship had a crew of 20 men, but 10 deserted before 
she sailed ; how many remained ? 10 from 20 leaves how 
many then ? Why ? 



SUBTRACTION. 


29 


Lessons 19 and 20. 

SUBTRACTION TABLE. 

Note. Questions in this table should not be asked in rotation, be¬ 
cause when they are so asked the learner can answer by merely count¬ 
ing, without the least exertion of memory. 

In one part of the table we find 3 from 8 leaves 5, and in another 
part, 5 from 8 leaves 3. Each operation in the table is repeated in this 
way, except when the number left is 1, or is the same as the number 
subtracted. Make the learner observe this. 


-2 

from 

3 

leaves 

i 

5 

from 

6 

leaves 

i 

8 

from 

9 

leaves 

i 

2 

from 

4 

leaves 

2 

5 

from 

7 

leaves 

2 

8 

from 

10 

leaves 

2 

2 

from 

5 

leaves 

3 

5 

from 

8 

leaves 

3 

8 

from 

11 

leaves 

3 

2 

from 

6 

leaves 

4 

o 

from 

9 

leaves 

4 

8 

from 

12 

leaves 

4 

2 

from 

7 

leaves 

5 

5 

from 

10 

leaves 

5 

8 

from 

13 

leaves 

5 

2 

from 

8 

leaves 

6 

o 

from 

11 

leaves 

6 

8 

from 

14 

leaves 

6 

2 

from 

9 

leaves 

7 

o 

from 

12 

leaves 

7 

8 

from 

15 

leaves 

7 

2 

from 

10 

leaves 

8 

5 

from 

13 

leaves 

8 

8 

from 

16 

leaves 

8 

2 

from 

11 

leaves 

9 

5 

from 

14 

leaves 

9 

8 

from 

17 

leaves 

9 

o 

from 

12 

leaves 

10 

5 

from 

15 

leaves 

10 

8 

from 

18 

leaves 

10 

3 

from 

4 

leaves 

1 

6 

from 

7 

leaves 

1 

9 

from 

10 

leaves 

1 

3 

from 

o 

leaves 

2 

6 

from 

8 

leaves 

2 

9 

from 

11 

leaves 

2 

3 

from 

6 

leaves 

3 

6 

from 

9 

leaves 

3 

9 

from 

12 

leaves 

3 

3 

from 

7 

leaves 

4 

6 

from 

10 

leaves 

4 

9 

from 

13 

leaves 

4 

3 

from 

8 

leaves 

5 

6 

from 

11 

leaves 

5 

9 

from 

14 

leaves 

5 

3 

from 

9 

leaves 

6 

6 

from 

12 

leaves 

6 

9 

from 

15 

leaves 

6 

3 

from 

10 

leaves 

7 

6 

from 

13 

leaves 

7 

9 

from 

16 

leaves 

7 

3 

from 

11 

leaves 

8 

6 

from 

14 

leaves 

8 

9 

from 

17 

leaves 

8 

3 

from 

12 

leaves 

9 

6 

from 

15 

leaves 

9 

9 

from 

18 

leaves 

9 

3 

from 

13 

leaves 

10 

6 

from 

16 

leaves 

10 

9 

from 

19 

leaves 

10 

4 

from 

5 

leaves 

1 

7 

from 

8 

leaves 

1 

10 

from 

11 

leaves 

1 

4 

from 

6 

leaves 

2 

7 

from 

9 

leaves 

2 

10 

from 

12 

leaves 

o 

*** 

4 

from 

7 

leaves 

3 

7 

from 

10 

leaves 

3 

10 

from 

13 

leaves 

3 

4 

from 

8 

leaves 

4 

7 

from 

11 

leaves 

4 

10 

from 

14 

leaves 

4 

4 

from 

9 

leaves 

5 

7 

from 

12 

leaves 

5 

10 

from 

15 

leaves 

5 

4 

from 

10 

leaves 

6 

7 

from 

13 

leaves 

6 

10 

from 

16 

leaves 

6 

4 

from 

11 

leaves 

7 

7 

from 

14 

leaves 

7 

10 

from 

17 

leaves 

7 

4 

from 

12 

leaves 

8 

7 

from 

15 

leaves 

8 

10 

from 

18 

leaves 

8 

4 

from 

13 

leaves 

9 

7 

from 

16 

leaves 

9 

10 

from 

19 

leaves 

9 

4 

from 

14 

leaves 

10 

7 

from 

17 

leaves 

10, 

10 

from 

20 

leaves 

10 


3* 






so 


SUBTRACTION. 


Lesson 21. 

To be performed in the mind. 

1. Augustus had 5 apples, but he soon after dropped 3 
and lost them ; how many had he left ? 

2. Benjamin has 6 cents and Samuel 4 ; which has the 
most ? How many the most ? 

3. If you have 7 pears, and give your sister 3 of them, 
how many will you have left ? 3 from 7 leaves how many 
then ? 4 from 7 ? 

4. I have 5 dollars ; how many more must I get to have 
8 ? 5 from 8 leaves how many then ? Why ? 

5. What is the difference between 4 and 8 ? 4 and 6 ? 
5 and 9 ? 4 and 7 ? 8 and 3 ? 8 and 6 ? 9 and 5 ? 10 and 
7 ? 6 and 12 ? 13 and 7 ? 15 and 8 ? 9 and 17 ? 

6. My father gave me 5 cents and my brother 4, but I 
soon after lost 3 of them ; how many did I then have ? 

7. A boy having 10 cents, spent 2 cents for some apples, 
and 6 cents for a little book; how many cents did he keep ? 

8. Julia has 16 plums; how many will be left if she gives 
away 4 ? 

9. If you take 7 dollars from 15, how many will be left ? 
How r many will be left if you take 3 from 11 ? 9 from 19 ? 
3 from 24 ? 6 from 33 ? 8 front 41 ? 5 from 55 ? 9 from 
62 ? 2 from 70 ? 4 from 85 ? 7 from 105 ? 3 from 120 ? 

10. George’s mother gave him 75 cents ; after he had 
spent 4 cents and 5 cents, how many had he left ? 

Lesson 22. 

To be performed in the mind. 

1. If you have 23 marbles, and lose 10, how many will 
you have left ? Explanation. 1 ten from 2 tens and 3 
leaves how many ? 

2. What remains after taking 10 apples from 15 ? 10 
from 19 ? 10 from 27 ? 10 from 32 ? 10 from 46 ? 10 from 
54 ? 10 from 63 ? 10 from 75 ? 10 from 81 ? 10 from 99 ? 
10 from 100 ? 10 from 108 ? 10 from 215 ? 10 from 455 ? 
10 from 1,000 ? 

3. A trader sold 20 pounds of sugar out of a box that 
contained 30 pounds ; how much was left ? Explanation. 
2 tens taken from 3 tens leave how many ? 


SUBTRACTION. 


SI 


4. A farmer set out 40 trees in an orchard, and his 
neighbor 70 ; which set out the most, the farmer or his 
neighbor ? How many the most ? 

5. What is the difference between 30 and 60 ? 20 and 
50 ? 60 and 80 ? 30 and 45 ? 40 and 78 ? 50 and 120 ; 
that is, between 5 tens and 12 tens ? 70 and 150 ? 

6. A grocer who had 20 cheeses sold 15 ; how many 
had he left ? Explanation. 5 from 20 leaves how many ? 
10 from 15 leaves how many ? 

7. If you have 40 cents, and pay away 25 of them for a 
book, how many will you have left ? 

8 A man who lives 16 miles from Boston, travelled 12 
miles towards that city ; how far was he from it then ? 

9. If you take 19 eggs from a basket that contains 32, 
how many will you leave ? 

10 . If you take 13 chestnuts from 18, how many will you 
leave ? How many will you leave if you take 21 from 37 ? 
32 from 40 ? 26 from 35 ? 57 from 72 ? 62 from 70 ? 200 
from 300 ? 150 from 230 ? 

Lesson 23. 

For the Slate. 

1. A young man having 665 dollars, paid 464 dollars for 
a piece of woodland ; how much money had he left ? 
operation. Explanation. We first write 464, 

6 6 5 the smaller number, under 665 the 

4 6 4 larger, with units under units, tens 

- under >tens, &c., and draw a line 

2 0 1 dollars. Ans. beneath. We now say 4 from 5 
leaves 1, we place this 1 under the 
units, and so proceed to take 6 tens from 6 tens, and 4 
hundreds from 6 hundreds ; thus, 6 from 6 leaves 0, 4 
from 6 leaves 2. 

The smaller number, which we subtract, is called the 
subtrahend , the. larger number the minuend , and the num¬ 
ber obtained, the difference or remainder; thus, in the pre¬ 
ceding example, 464 is the subtrahend, 665 the minuend, 
and 201 the difference or remainder. 


How do we proceed in example 1, lesson 23, to subtract 464 from 665 P 
What is called the subtrahend ? Minuend ? Difference or remainder ? 
In example 1, lesson 23, what is the subtrahend ? Minuend ? Difference 
or remainder ? 




32 


SUBTRACTION. 


2. A planter, who had 96 bales of cotton, sent 45 of them 

to Charleston ; how many were left ? Ans. 51. 

3. If you have 688 mill-logs, and get 133 of them sawed, 

what number will remain ? Ans. 555. 

4. A man who had 45,647 feet of boards, sent away 
21,546 feet; what quantity did he keep ? Ans. 24,101 feet. 

5. A merchant, worth 8,'946 dollars, lost 805 dollars in a 
speculation ^ how much was he then worth ? 

Ans. 8,141 dollars. 

6. A farmer in Michigan raised 1,507 bushels of wheat, 
and sold 1,103 bushels ; what quantity did he retain ? 

Ans. 404 bushels. 


Lesson 24. 

1. A man who had 43 acres of land, sold 26 acres ; how 
many acres did he keep ? 

operation. Explanation . Being unable to take 

4 3 6 units from 3, we borrow 1 of the 4 

2 6 tens, add it to 3, and take 6 from 13 ; 

- having now only 3 tens left in the up- 

1 7 acres. Ans. per number, we take 2 tens from 3 tens. 

Another better loay. Being unable to take 6 units from 
3, we add 10 to 3, and take 6 from 13 ; as we have added 
1 ten to the upper number, to balance it, we now add 1 ten 
to the lower number, and take 3 tens from 4 tens. We 
proceed in subtracting thus ; 6 from 13 leaves 7, 1 carried 
to 2 makes 3, 3 from 4 leaves 1. 

2. A farmer raised 636 bushels of potatoes, and after 

saving 457 bushels for himself, sold the remainder ; how 
many bushels did he sell ? Ans. 179. 

Explanation. Add 1 hundred, or 10 tens, to 3 tens, in 
order to subtract, and then carry 1 hundred to the 4 
hundred. 

3. A ship had 35 persons on board, including the pas¬ 

sengers and crew ; how r many passengers were there if 
the crew consisted of 9 men ? Ans. 26. 

4. If you have 8,005 dollars, and buy a farm for 5,126 
dollars, how much money will you have left ? 

Ans. 2,879 dollars. 


Explain how we subtract 26 from 43. in example 1, lesson 24. How 
do we proceed in subtracting 26 from 43 ? 





SUBTRACTION. 


33 


5. A company of merchants commenced trade with one 
hundred and twelve thousand three hundred and five dol¬ 
lars ; at the end of a year they had eighty-one thousand 
three hundred and seven dollars remaining ; how much 
had they lost ? Ans. 30,998 dollars. 

Lesson 25. 

From what precedes we get the following 

RULE FOR SUBTRACTION. 

Write the smaller number under the larger, icith units under 
units, tens under tens, fyc., and draw a line beneath. Sub¬ 
tract the units in the lower number from the figure above, and 
put the remainder directly below; proceed in the same way 
with the tens, hundreds, fyc. If a figure in the lower num¬ 
ber be larger than the figure above, add 10 to the one above , 
subtract, cm.d then carry 1 to the next figure or place below. 

If you take 3 walnuts from a heap that contains 8, there 
will be 5 left. If you now add the 3 walnuts to the re¬ 
mainder 5, there will be 8 in the heap again. 

Therefore, as we sometimes make a mistake in subtract¬ 
ing, we can prove the work. 

By adding the smaller number to the remainder; if the 
sum be equal to the larger number, the work will generally 
be right. 

1. Subtract 673 from 1,782. 

OPERATION. PROOF. 

1782 To 1109 remainder, 

6 7 3 Add 6 7 3 smaller number, 

1,1 0 9 remainder. 1,7 8 2 same as the greater number. 

JVoie. Each example should now be proved. 

(2.) (3.) 

From 6 0 0 2 5 From 324573819746 

take 4 6 take 147233518869 


How do you write the numbers in Subtraction ? What do you sub¬ 
tract first? Where do you put the remainder? How then do you pro¬ 
ceed ? What if a figure in the lower number be larger than the figure 
above ? 

If you take 3 walnuts from a heap that contains 8, how many will 
be left ? If you now add the 3 walnuts to the remainder 5, how many will 
there be in the heap again? 

How, then, can we prove the work in Subtraction ? 







34 


SUBTRACTION. 


(4.) (5.) 

From 5 1 0 0 0 2 From 1 7 4 8 3 3 

take 1 2 0 0 0 5 take 5 2 0 0 3 


Lesson 26. 

1. From 64,321,535,488,327 take 10,035,780,321,199. 

2. From 9,000,052 take 1,643,215. 

3. From 750,081 take 3,011. 

4. 81 men were engaged in quarrying stone ; after a 

number were discharged, there were 33 left ; how many 
were discharged ? Ans. 48. 

5. George Washington died in the year 1799, aged 67 

years ; in what year was he born ? Ans. in 1732. 

6. A man who had 11,375 dollars in his possession, paid 

all his debts, amounting to 3,287 dollars; how much money 
had he left ? Ans. 8,088 dollars. 

7. By how much do 48,323 dollars exceed 19,295 dol¬ 
lars ? Ans. 29,028 dollars. 

8. Abrickmaker had a kiln containing 43,000 bricks ; 
after he had sold 25,376, how many remained ? Ans. 17,624. 

9. If you buy 275 bushels of wheat for 400 dollars, and 
sell 125 bushels for 198 dollars, how many bushels have 
you left, and what have they cost you ? 

Ans., you have 150 bushels left, and they have cost you 
202 dollars. 

10. If you have 23 apples, and a boy gives you 16 more, 

how many will you have after giving your sister 15, and 
your brother 8 ? Ans. 16. 

Lesson 27. 

1. A merchant who had 5,635 pounds of coffee, sold 

1,210 pounds to one man, and 3,145 to another; how much 
had he then left ? Ans. 1,280 pounds. 

2. A tree seventy-five feet high, was broken off* by the 

wind, and the part that fell was thirty-nine feet long ; how 
high was the stump left standing ? Ans. 36 feet. 

3. A grocer sold a quantity of butter for 103 dollars, 
and made 27 dollars on it ; how much did it cost him ? 

Ans. 76 dollars. 

4. How many years is it since the Pilgrims landed at 
Plymouth, in the year 1620 ? 

5. If a clerk collects 2,750 dollars from one person, and 
1,385 from another, and pays out of it 275 dollars to one 
man, and 323 to another, how much will be left ? 

Ans. 3,537 dollars. 





MULTIPLICATION. 


35 


6. In 1830 France contained 32,052,465 inhabitants, and 
the British Empire 22,297,621 ; how many more people 
had France than the British Empire ? Ans. 9,754,844. 

7. If I travel 2,332 miles from home, and then return 
1,421 miles, how far am I from home then ? Ans. 911 miles. 

8. A man gave 15,235 dollars for a ship, and sold her 
for 12,250 dollars, how much did he lose ? 

Ans. 2,985 dollars. 

9. If you buy a horse and chaise for 235 dollars, and 

sell the chaise for 147 dollars, what will be the cost of the 
horse to you ? Ans. 88 dollars. 

10. There is a bin that holds 2,000 bushels ; if there are 
9 bushels of corn in it, how many more will it hold ? 

Ans. 1,991 bushels. 


MULTIPLICATION. 

Lesson 28. 

To be performed in the mind. 

1. If one orange costs 3 cents, what will 2 oranges cost? 
2 times 3 are how many then ? 3 times 2 are how many ? 

2. Mary recites 3 lessons a day ; how many will she 
recite in 5 days ? 

3. A merchant bought 4 barrels of flour at 6 dollars a 
barrel ; what sum must he pay for the whole ? What 
number then are 4 times 6 ? 6 times 4 ? 

4. What shall I pay for 6 yards of cloth, at 2 dollars a 
yard ? 

5. What sum must you pay for 7 weeks’ board, at 3 dol¬ 
lars a week ? 

6. A man walked 4 miles an hour for 8 hours ; how far 
did he go in that time ? 8 times 4 are how many then ? 4 
times 8 ? 

7. George bought 5 camels’ hair pencils at 5 cents 
apiece ; what did he pay for them ? 

8. A gardener gave 9 children 2 apples apiece ; what 
number did they all receive ? 

9. How many days are there in 10 weeks, there being 
7 days in a week ? 

10. What sum must I give for 6 pounds of sugar at 8 
cents a pound ? How many are 6 times 8 then ? 8 times 6 ? 



36 


MULTIPLICATION. 


Lessons 29 and 30. 


MULTIPLICATION TABLE. 

In one part of the table we find 4 times 7 are 28, and in another 
part, 7 times 4 are 28. Each operation in the table is repeated in this 
way, except when a number is multiplied by 1, or itself. Make the 
learner observe this. 


2 

times 

i 

are 

2 

5 

times 

i 

are 

5 

8 

times 

1 

are 

8 

2 

times 

2 

are 

4 

5 

times 

2 

are 

10 

8 

times 

C 1 

** 

are 

16 

2 

times 

3 

are 

6 

5 

times 

3 

are 

15 

8 

times 

3 

are 

24 

2 

times 

4 

are 

8 

5 

times 

4 

are 

20 

8 

times 

4 

are 

32 

°2 

times 

5 

are 

10 

5 

times 

5 

are 

25 

8 

times 

5 

are 

40 

2 

times 

6 

are 

12 

5 

times 

6 

are 

30 

8 

times 

6 

are 

48 

2 

times 

7 

are 

14 

5 

times 

7 

are 

35 

8 

times 

7 

are 

56 

2 

times 

8 

are 

16 

5 

times 

8 

are 

40 

8 

times 

8 

are 

64 

2 

times 

9 

are 

18 

5 

times 

9 

are 

45 

8 

times 

9 

are 

72 

2 

times 

10 

are 

20 

5 

times 

10 

are 

50 

8 

times 

10 

are 

80 

3 

times 

1 

are 

3 

6 

times 

1 

are 

6 

9 

times 

1 

are 

9 

3 

times 

2 

are 

6 

6 

times 

2 

are 

12 

9 

times 

2 

are 

18 

3 

times 

3 

are 

9 

6 

times 

3 

are 

18 

9 

times 

3 

are 

27 

3 

times 

4 

are 

12 

6 

times 

4 

are 

24 

9 

times 

4 

are 

36 

3 

times 

5 

are 

15 

6 

times 

5 

are 

30 

9 

times 

5 

are 

45 

3 

times 

6 

are 

18 

6 

times 

6 

are 

36 

9 

times 

6 

are 

54 

3 

times 

7 

are 

21 

6 

times 

7 

are 

42 

9 

times 

7 

are 

63 

3 

times 

8 

are 

24 

6 

times 

8 

are 

48 

9 

times 

8 

are 

72 

3 

times 

9 

are 

27 

6 

times 

9 

are 

54 

9 

times 

9 

are 

81 

3 

times 

10 

are 

30 

6 

times 

10 

are 

60 

9 

times 

10 

are 

90 

4 

times 

1 

are 

4 

7 

times 

1 

are 

7 

10 

times 

1 

are 

10 

4 

times 

2 

are 

8 

7 

times 

2 

are 

14 

10 

times 

2 

are 

20 

4 

times 

3 

are 

12 

7 

times 

3 

are 

21 

10 

times 

3 

are 

30 

4 

times 

4 

are 

16 

7 

times 

4 

are 

28 

10 

times 

4 

are 

40 

4 

times 

5 

are 

20 

7 

times 

5 

are 

35 

10 

times 

5 

are 

50 

4 

times 

6 

are 

24 

7 

times 

6 

are 

42 

10 

times 

6 

are 

60 

4 

times 

7 

are 

28 

7 

times 

7 

are 

49 

10 

times 

7 

are 

70 

4 

times 

8 

are 

32 

7 

times 

8 

are 

56 

10 

times 

8 

are 

80 

4 

times 

9 

are 

36 

7 

times 

9 

are 

63 

10 

times 

9 

are 

90 

4 

times 

10 

are 

40 

7 

times 

10 

are 

70 

10 

times 

10 

are 

100 














MULTIPLICATION. 


37 


Lesson 31. 

To be 'performed in the mind. 

1. Alfred has 3 little boxes, in each of which he has 4 
cents ; how many cents has he ? 

2. Nathan has 6 plums, and Robert 5 times as many ; 
how many has Robert ? 

3. A man bought 7 sheep, at 4 dollars apiece ; how 
many dollars did he give for all of them ? 

4. If a picture book costs 9 cents, what will 6 such books 
cost ? How many then are 6 times 9 ? 9 times 6 ? 

5. How many are 3 times 2 ? 2 times 4 ? 2 times 9 ? 
3 times 7 ? 5 times 4 ? 8 times 3 ? 5 times 6 ? 6 times 6 ? 
7 times 4 ? 8 times 9 ? 9 times 6 ? 9 times 9 ? 

6. A farmer bought 3 pigs of one man, and 4 of another, 
paying 3 dollars apiece for them ; what did they all cost 
him ? 

7. A boy had 2 melons, which he sold at 10 cents apiece; 
what sum did he get for them ? How many are 3 times 
10 ? 5 times 10 ? 4 times 10 ? 8 times 10 ? 7 times 10 ? 
6 times 10 ? 9 times 10 ? 

8. There are 2 rows of plum trees in a garden, with 20 
trees in a row ; how many trees are there in both rows ? 

9. If you buy 6 books at 40 cents apiece, how many 
cents must you pay for them ? Explanation. 6 times 4 
tens are how many tens ? What sum do 10 tens make ? 
What sum do 20 tens, that is, 2 times 10 tens make ? What 
sum then do 2 hundred and 4 tens make ? 

10. How many are 2 times 30 ? 2 times 50 ? 3 times 
20 ? 4 times 30 ? 6 times 50 ? 5 times 80 ? 7 times 20 ? 

9 times 30 ? 2 times 200 ? 4 times 400 ? 

Lesson 32. 

To be performed in the mind. 

1. A laborer earned 2 dollars a day for 10 days ; how 
much did his wages amount to in that time ? 

2. How many are 10 times 3 ? 10 times 6 ? 10 times 4 ? 

10 times 5 ? 10 times 9 ? 10 times 7 ? 10 times 8 ? 10 
times 10 ? 

3. Joseph has 20 filberts, and Daniel 10 times as many; 
what number has Daniel ? 

4 


38 


MULTIPLICATION. 


4. How many are 10 times 30 ? 10 times 40 ? 10 times 
70 ? 10 times 80 ? 10 times 60 ? 10 times 50 ? 10 times 
90 ? 10 times 100 ? 10 times 25 ? 10 times 44 ? 

5. 2 men were paid 16 dollars each ; what sum did they 
both receive ? Explanation. 2 times 6 are how many ? 2 
times 1 ten are how many tens ? How many, then, are 2 
tens and 12, or 2 tens, 1 ten, and 2 ? 

6. If a team hauls 32 bushels of corn in one load, how 
many bushels can it haul in 2 loads ? 

7. How many lines are there in 5 pages of your book, 
if there are 15 lines in a page ? 

8. A boy has 4 25-cent pieces ; how many cents are 
they worth ? 

9. There are 63 gallons in a hogshead ; how many gal¬ 
lons are there in 2 hogsheads ? 

10. How many are 3 times 11 ? 2 times 12 ? 4 times 
21 ? 3 times 38 ? 5 times 22 ? 6 times 13 ? 2 times 91 ? 
8 times 42 ? 


Lesson 33. 


OPERATION. 

352 

4 

1,4 0 8 dollars. Ans. 


For the Slate . 

1. 4 sons inherited 352 dollars apiece ; how much did 
they all inherit ? 

Explanation. We first write the 
smaller number under the larger, 
with units under units, and draw a 
line beneath. We now say 4 times 
2 are 8 ; and place the 8 under the 
units ; then 4 times 5 tens are 20 
tens, or 2 hundred. There being no odd tens, we put 0 in 
the tens’ place, carry the 2 hundred one place to the left, 
and add it to the 12 hundreds which are obtained by mul¬ 
tiplying the 3 hundreds by 4, thereby making 14 hundred, 
or 1 thousand 4 hundred. The 4 hundred we put in the 
hundreds’ place, and the 1 thousand we carry to the thou¬ 
sands’ place. 

The lower number, which we multiply by, is called the 
multiplier , the upper number, which we multiply, the mul¬ 
tiplicand, and the number produced the product; thus, in 


In example 1, lesson 33, how do we proceed to multiply 352 by 4 ? 
What is called the multiplier P Multiplicand ? Product? 




MULTIPLICATION. 


39 


the preceding example, 4 is the multiplier, 352 the multi¬ 
plicand, and 1,408 the product. 

2. How many pounds are there in 15 2-pound weights ? 

Ans. 30. 

3. How much flour is there in 9 barrels, each of which 

contains 196 pounds ? Ans. 1,764 pounds. 

4. There are 5,280 feet in one mile, and 3 miles in one 
league ; now how many feet are there in 6 leagues. 

Ans. 95,040. 

5. Suppose that there are one hundred and eighty-five 
millions of people in Europe, and that the Earth contains 
four times as many ? how many are there on the Earth ? 

Ans. 740,000,000. 


Lesson 34. 


1. How much will it cost to build a road 127 miles long, 
at 705 dollars a mile ? 

Explanation. We multiply first 
by the 7 units, then by the 2 tens, 
and as the product is ten times as 
much as it would have been had 
we multiplied by 2 units, we put it 
beneath the first product, one place 
to the left ; afterwards we multiply 
by the 1 hundred, and as the pro¬ 
duct is a hundred times as much as 
it would have been had we multi- 


OPERATION. 

7 05 
1 2 7 

4 935 
14 10 
7 05 


8 9,5 3 5 dollars. Ans. 


plied by 1 unit, we put it beneath the other products two 
places to the left; finally, adding up the three products, we 
get the whole product. 

This is plainly the answer, since we have taken 705, 
7, 20, and 100 times, and have added the products together. 

Observe that the right hand figure of each product is 
placed directly under the figure we multiply by. 

2: How much are 134 tons of hay worth, at 25 dollars a 
ton ? Ans. 3,350 dollars. 

3. What sum must I pay for 2,327 chaldrons of orrel 

coal, at 14 dollars a chaldron ? Ans. 32,578 dollars. 

4. How many gallons of molasses have I in three hun- 


In example 1, lesson 33, what is the multiplier ? Multiplicand ? Pro¬ 
duct ? 

How do we proceed to multiply 705 by 127 in example 1, lesson 34? 
Where is the right hand figure of each product placed ? 





40 


MULTIPLICATION. 


dred and forty-five hogsheads, each hogshead containing 
ninety-four gallons ? Ans. 32,430. 

5. If 112 emigrants to the West are provided with 1,235 
dollars apiece, how much money have they all ? 

Ans. 138,320 dollars. 


Lesson 35. 

From what precedes we get the following 

RULE FOR MULTIPLICATION. 

Write one number under the other, with units under 
units, tens under tens, fyc., and draw a line beneath. Then 
beginning at the right of the multiplicand , multiply each 
figure in it by the units in the multiplier , carry as in Addi¬ 
tion, and write the product below. If the multiplier contain 
but one figure, the operation is now done. If it contain more 
than one figure, multiply in the same manner by the tens, hun¬ 
dreds, fyc., in the multiplier, taking care to put the first 
figure of each product directly beneath the figure by which 
you multiply, and finish by adding up the several products. 

Note. It is generally best to make the smaller number the multiplier. 

5 times 9 are 45, and 9 times 5 are 45. 

Therefore, as we sometimes make a mistake in multiply¬ 
ing, we can prove the work, 

By multiplying the former multiplier by the multiplicand ; 
if the product be equal to the first, the work will generally be 
right. 

1. Multiply 412 by 18. 


OPERATION. 

PROOF 

4 1 2 

1 8 

1 8 

4 1 2 

3 2 9 6 

3 6 

4 1 2 

1 8 

7,4 1 6 product. 

7 2 


7,4 1 6 product, same as before. 


How do you write the numbers in Multiplication ? llow then do you 
multiply, carry, and write the product P What if the multiplier contain 
but one figure? What if the multiplier contain more than one figure -v 
What number should generally be made the multiplier ? 

How many are 5 times 9 ? 9 times 5 ? 

How then can we prove the work in Multiplication? 







MULTIPLICATION. 


41 


Note. Each example should now be proved. 

(2.) (3.) (4.) 

Multiply 6 3 5 7 Multiply 4 8 2 1 Multiply 4 3 0 0 2 
by 2 3 3 5 by 113 1 by 2 5 


•Lesson 36. 

1. Multiply 254,420,335 by 3,347,889. 

2. Multiply 815,555 by 5,542. 

3. Multiply 5,001 by 357. 

4. How much must I give for 234 hogsheads of molasses, 

at 19 dollars a hogshead ? Ans. 4,446 dollars. 

5. 8 men trading in company, furnished 5,237 dollars 
apiece ; how much did they all furnish ? 

Ans. 41,896 dollars. 

6. I have a book which contains 421 pages, and each 
page 42 lines ; how many lines are there in the book ? 

Ans. 17,682. 

7. What number will you get if you take 646,325, 5,335 

times ? Ans. 3,448,143,875. 

8. What must I give for 35 yards of broadcloth, at 5 

dollars a yard ? Ans. 175 dollars. 

9. How much must a man give for twenty-five barrels 
of flour, at twelve dollars a barrel ? Ans. 300 dollars. 

10. A certain army contains 13 regiments, with 832 men 
in a regiment; how many men are there in the whole army ? 

Ans. 10,816. 


Lesson 37. 

1. What is the product of twelve thousand two hundred 
and twelve by three hundred and seventy-five ? 

Ans. 4,579,500. 

2. 225 men did a piece of work in 313 days ; how long 
would it have taken one man to do it ? Ans. 70,425 days. 

3. How much will it cost to build a rail-road 26 miles 

long, at 8,231 dollars a mile ? Ans. 214,006 dollars. 

4. If you buy 473 tons of iron, at 116 dollars a ton, how 

much must you pay for it ? Ans. 54,868 dollars. 

5. A man hired 12 persons to labor for him at 1 dollar a 

day, apiece ; how much did their wages amount to in 303 
days ? Ans. 3,636 dollars. 

6. If a chaise wheel turns round 351 times in a mile, 

4* 





42 


MULTIPLICATION. 


how many times will it turn round in going from Boston to 
Providence, 41 miles ? Ans. 14,391 times, 

7. How much must I pay for 9 cows, at 18 dollars 

apiece ? Ans. 162 dollars. 

8. Suppose a steamboat between Providence and New 

York moves 15 miles an hour ; how far will she go at this 
rate in 12 hours ? * Ans. 180 miles, 

9. What sum must a man pay for 459 barrels of cider, 

at 3 dollars a barrel ? Ans. 1,377 dollars. 

10. 13 men received 189 dollars apiece ; how much did 

they all receive ? Ans. 2,457 dollars. 


Lesson 38. 


1. There is a book that has 21 pages, each page contains 
16 lines, and each line 8 words; how many words are there 
in the book ? 

OPERATION, 

2 1 pages. 

1 6 lines in a page. 


OR 

1 6 lines in a page, 
8 words in a line. 


1 2 6 
2 1 


3 3 6 lines in the book. 
8 words in a line. 


12 8 words in a page. 
2 1 pages. 


12 8 
256 


2,6 8 8 words. Ans. 2,6 8 8 words. Ans. 

Therefore, when several numbers are to be multiplied 
together to obtain an answer, 

TVe can multiply the numbers together in any order we 
please , and the answer will always be the same. 

2. If I hire 35 men at 4 dollars a week, apiece, what 

sum will their wages amount to in 3 years, there being 52 
weeks in a year ? Ans. 21,840 dollars. 

3. Multiply 7, 19, 27, 236, and 11 together. 

Ans. 9,322,236. 

4. Multiply 33, 5,321, and 424 together. 

Ans. 74,451,432. 

5. How many bushels of potatoes are there in 12 loads, 
each load containing 10 barrels, and each barrel 3 bushels ? 

Ans. 360 bushels. 


When several numbers are to be multiplied together to obtain an 
tuswer, how can we multiply them ? 










multiplication. 


43 


6. It a vessel sail 9 miles an hour, how far will she sail 

in a week ; there being 24 hours in a day, and 7 days in a 
week? Ans. 1,512 miles. 

7. What is the product of 5, 6, 7, 8, 9, and 25 multi¬ 
plied together ? Ans. 378,000. 

CONTRACTIONS IxN MULTIPLICATION. 

Lesson 39. 

1. If I buy 103 pieces of cloth, at 256 dollars apiece, 
how much do I pay for the whole ? 

OPERATION. 

2 5 6 
1 0 3 


7 6 8 
25 6 


2 6,3 6 8 dollars. Ans. 

Os therefore , between the figures of the multiplier, are not 
used. 

2. Multiply 245,181 by 6,005 ? Ans. 1,472,311,905. 

3. How many bricks did a team haul away from a brick¬ 
yard in 206 loads, each load containing 1,296 bricks ? 

Ans. 266,976. 

4. Multiply 3,837 by 2,108. Ans. 8,088,396. 

5. If a cow is worth 25 dollars, what are 10 such cows 

worth ? Ans. 250 dollars. 

Erplanation. Put 0 at the right of 25, and it increases 
its value 10 times, or multiplies it by 10. See Numera¬ 
tion, latter part of lesson 5. 

6. What must I give for 4 horses, at 100 dollars apiece ? 

Ans. 400 dollars. 

7. If 1,000 men are paid 165 dollars apiece, how much 

are the whole paid ? Ans. 165,000 dollars. 

So when two numbers are to be multiplied together, if 
one is 10, 100, &c. 

We put the Os in it at the right of the other to get the product. 


What is done with the Os between the figures of the multiplier? 
Explain how example 5, lesson 39, is performed. 

What is done when two numbers are to be multiplied together, if on® 
is 10, 100, &c.? 





44 


MULTIPLICATION. 


8. 1,000 men before going into battle, were furnished with 
45 cartridges apiece; how many cartridges did they receive ? 

9. There being 100 cents in a dollar, how many cents 
are there in 3 dollars ? 

10. What is the product of 1,000 by 255. 

Lesson 40. 

1. If a vessel sail 120 miles a day for 6 days, how far 
will she go in that time ? 

operation. j Explanation. We here multiply 12 

12 0 tens by 6, and put the 0 in 120 at the 

6 right of the product to show that it is 

- 72 tens. 

7 2 0 miles. Ans. 

2. A drover has 300 cows worth 25 dollars apiece ; 
what is the value of all of them ? 

operation. Explanation. 25 taken 3 hun- 

2 5 dred times, or which is the same 

3 0 0 thing, 3 hundred taken 25 times, 

- makes 75 hundred ; we therefore 

7,5 0 0 dollars. Ans. put the 0s in 300 at the right of 75. 

3. If 200 men have 150 dollars apiece, how much have 
they all ? 

operation. Explanation. 15 tens taken 2 

15 0 times make 30 tens; we therefore 

2 0 0 put the 0 in 150 at the right of 

- 30, and as 2 hundred times 15 

3 0,0 0 0 dollars. Ans. tens are a hundred times as much 

as 2 times 15 tens, we put the 
two 0s in 200 at the right of 300. 

Therefore, when there are 0s at the right of the multi¬ 
plier, or multiplicand, or both, 

Omit them entirely in multiplying , hut place them at the 
right of the product. 

4. What quantity of shad are there in 65 barrels, each 

of which contains 200 pounds ? Ans. 13,000 pounds. 

5. Multiply 432,000 by 2,100. Ans. 907,200,000 


Explain how example 1, lesson 40, is performed. 

Explain how example 2, lesson 40, is performed. 

Explain how example 3, lesson 40, is performed. 

What course do we take when there are 0s at the right of the multi* 
plier, or multiplicand, or both ? 






DIVISION. 


45 

6. What sum are 160 hogsheads of molasses worth, at 

20 dollars a hogshead ? Ans. 3,200 dollars. 

7. A company of men bought 6 acres of land at 2,000 
dollars an acre ; how much did they give for it ? 

Ans. 12,000 dollars. 

8. How much must a merchant pay for 250 bales of 

cotton, at 80 dollars a bale ? Ans. 20,000 dollars. 

9. What is the product of eighty-three thousand five 
hundred, and nine hundred and seventy-seven ? 

Ans. 81,579,500. 

10. If one ton of hay is worth 20 dollars, what are 9 tons 

worth ? Ans. 180 dollars. 


DIVISION. 

Lesson 41 . 

To be performed in the mind. 

1. How many apples can you get for 6 cents, at 2 cents 
apiece ? 2 is in 6 how many times then ? 3 is in 6 how 
many times ? 

2. Emma had 9 roses, which she divided equally among 
3 children ; how many did she-give to each ? 

3. I rode 15 miles in 3 hours ; how far did I ride in one 
hour ? How many times is 3 in 15 then ? Why is 3 in 15 
5 times ? Answer. Because 3 times 5 are 15. 

4. A trader paid 16 dollars for 4 hats ? what was the 
price of one hat ? 4 is contained in 16 how many times 
then ? Why ? 

5. If you have 18 sugar plums, 6 times as many as 
Charles, how many has Charles ? 

6. 7 boys have 21 pears ; if they divide them equally, 
what will be the share of each ? 

7. How many hours will a traveller be in going 40 miles, 
if he rides 8 miles an hour ? What number then does 40 
divided by 8 give ? 40 divided by'5 ? 

8. A man earned 54 dollars in 9 weeks ; how much did 
he earn in one week ? 

9. If you divide 6J9 cents into 10 equal heaps, how many 
will there be in each heap ? 

10. 6 men did a piece of work for 42 dollars : what was 
each one's share of the pay ? How many times then is 6 
in 42 ? Why ? 



46 


DIVISION. 


\ 


Lessons 42 and 43. 

\ 


DIVISION TABLE. 


Note. Questions in this table should not be asked in rotation, be¬ 
cause when they are so asked the learner can answer by merely count¬ 
ing, without the least exertion of memory. 

In one part of the table we find 5 in 35, 7 times, and in another part, 
7 in 35, 5 times. Each operation in the table is repeated, in this way, 
except when a number is divided by itself, or is contained in another as 
many times as 1 is in itself. Make the learner observe this. 


2 

in 

2, 

1 

time 

5 

in 

5, 

2 

in 

4, 

2 

times 

5 

in 

10, 

2 

in 

6, 

3 

times 

5 

in 

15, 

2 

in 

8, 

4 

times 

5 

in 

20, 

2 

in 

10, 

5 

times 

5 

in 

25, 

2 

in 

12, 

6 

times 

5 

in 

30, 

2 

in 

14, 

7 

times 

5 in 

35, 

2 

in 

16, 

8 

times 

5 4n 

40, 

2 

in 

18, 

9 

times 

5 

in 

45, 

2 

in 

20, 

10 

times 

5 

in 

50, 

3 

in 

3, 

1 

time 

6 

in 

6, 

3 

in 

6, 

2 

times 

6 

in 

12, 

3 

in 

9, 

3 

times 

6 

in 

18, 

3 

in 

12, 

4 

times 

6 

in 

24, 

3 

in 

15, 

5 

times 

6 

in 

30, 

3 

in 

18, 

6 

times 

6 

in 

36, 

3 

in 

21, 

7 

times 

6 

in 

42, 

3 

in 

24, 

8 

times 

6 

in 

'48, 

3 

in 

27, 

9 

times 

6 

in 

54, 

3 

in 

30, 

10 

times 

6 

in 

60, 

4 

in 

4, 

1 

time 

7 

in 

7, 

4 

in 

8, 

2 

times 

7 

in 

14, 

4 

in 

12, 

3 

times 

7 

in 

21, 

4 

in 

16, 

4 

times 

7 

in 

28, 

4 

in 

20, 

5 

times 

7 

in 

35, 

4 

in 

24, 

6 

times 

7 

in 

42, 

4 

in 

28, 

7 

times 

7 

in 

49, 

4 

in 

32, 

8 

times 

r* 

4 

in 

56, 

4 

in 

36, 

9 

times 

7 

in 

63, 

4 

in 

40, 

10 

times 

7 

in 

70, 


1 

time 

8 

in 

8, 

1 

time 

2 

times 

8 

in 

16, 

2 

times 

3 

times • 

8 

in 

24, 

3 

times 

4 

times 

8 

in 

32, 

4 

times 

5 

times 

8 

in 

40, 

5 

times 

6 

times 

8 

in 

48, 

6 

times 

7 

times 

8 

in 

56, 

7 

times 

8 

times 

8 

in 

64, 

8 

times 

9 

times 

8 

in 

72, 

9 

times 

10 

times 

8 

in 

80, 

10 

times 

1 

time 

9 

in 

S', 

1 

time 

2 

times 

9 

in 

18, 

2 

times 

3 

times 

9 

in 

27, 

3 

times 

4 

times 

9 

in 

36, 

4 

times 

5 

times 

9 

in 

45, 

5 

times 

6 

times 

9 

in 

54, 

6 

times 

7 

times 

9 

in 

63, 

7 

times 

8 

times 

9 

in 

72, 

8 

times 

9 

times 

9 

in 

81, 

9 

times 

10 

times 

9 

in 

90, 

% 

10 

times 

1 

time 

10 

in 

10, 

1 

time 

2 

times 

10 

in 

20, 

2 

times 

3 

times 

10 

in 

30, 

3 

times 

4 

times 

10 

in 

40, 

4 

times 

5 

times 

10 

in 

50, 

5 

times 

6 

times 

10 

in 

60, 

6 

times 

7 

times 

10 

in 

70, 

7 

times 

8 

times 

10 

in 

80, 

8 

times 

9 

times 

10 

in 

90, 

9 

times 

10 

times 

10 in 100, 

10 

times 












DIVISION. 


47 


Lesson 44. 

To be performed in the mind. 

1. William had 8 cents, which he laid on the table in 2 
equal heaps ; how many were there in each heap ? If he 
had had 7 cents, how many would there have been in each 
heap, and how many over ? 

2. A man received 12 dollars for 4 days’ labor ; what 
were his wages a day ? 

3. How many heaps of 4 cents in a heap can you make 
out of 14 cents, and how many will there be over ? 

4. If you divide 32 apples equally among 8 boys, how 
many will each receive ? How many times then is 8 in 32 ? 
4 in 32 ? 

5. How many times is 2 in 12 ? 3 in 27 ? 4 in 28 ? 4 
in 36 ? 5 in 40 ? 7 in 28 ? 6 in 54 ? 6 in 36 ? 8 in 32 ? 
8 in 64 ? 9 in 36 ? 10 in 80 ? 

6. If you have 43 pounds of butter, and sell it in parcels 
of 5 pounds each, how many of such parcels will there be, 
and how many pounds over ? 

7. 2 men had equal shares in 20 dollars ; what was 
each one’s portion ? 

8. 30 cents were equally divided among 2 boys ; how 
many did each one receive ? Explanation. 30 is 3 tens ; 
how many tens did each of the 2 boys receive, and how 
many cents were there over ? How many times is 2 in 10 ? 
How many then are 1 ten and 5 ? 

9. How many times is 2 contained in 24 ? In 26 ? In 
32 ? In 46 ? In 60 ? In 48 ? In 82 ? In 100 ? 

10. A man bought 3 pounds of butter for 42 cents; what 
was the price a pound ? What would the price have been 
a pound if he had given 39 cents ? 

Lesson 45. 

To be performed in the mind. 

1. How many times is 3 contained in 30 ? In 36 ? In 
48 ? In 54 ? In 60 ? In 75 ? In 90 ? 

2. How many yards of broadcloth do I get for 48 dollars, 
if I pay 4 dollars a yard ? 

3. How many times is 4 contained in 40 ? In 52 ? In 
64 ? In 80 ? In 96 ? In 100 ? 

4. How many times is 5 contained in 55 ? 5 in 70 ? 6 in 
78 ? 6 in 96 ? 7 in 70 ? 7 in 91 ? 8 in 96 ? 9 in 99 ? 


48 


DIVISION. 


5. 2 shipwrecked sailors divided 110 biscuits equally 
between them ; what was each one’s share ? Explanation 
110 is 11 tens ; now how many tens did each receive, and 
how many were there over ? How many times is 2 contain¬ 
ed in 10 ? How many then are 5 tens and 5 ? 

6. How many barrels of apples can you buy for 120 
dollars, at 3 dollars a barrel ? 

7. How many times is 2 contained in 200 ? 3 in 300 ? 
4 in 400 ? 4 in 440 ? 5 in 600 ? 7 in 700 ? 9 in 540 ? 

8. If you give 30 cents for 10 oranges, what is the price 
of one orange ? 

9. How many times is 10 contained in 40 ? In 60 ? 
In 90 ? In 120 ? In 160 ? In 170 ? In 200 ? In 300 ? In 
370 ? In 450 ? In 540 ? 

10. How many times is 2 contained in 25, and what is 

left ? How many times is 3 contained in 38, and what is 

left ? How many times is 5 contained in 67, and what is 

left ? How many times is 8 contained in 100 and what is 

left ? How many times is 10 contained in 96, and what 
is left ? 

Lesson 46. 

For the Slate. 

Way of dividing when the number we divide by consists 
of only one figure. 

1. A merchant having 527 dollars in silver, wished to 
get the amount in 2-dollar bills ; how many of such bills 
ought he to receive in exchange ? 

operation. Explanation. We place 2 at 

2)5 2 7 the left of 527, separate them by 

- a curved line, and draw a line 

2 6 3 1 remainder, beneath. Now 2 is in the 5 hun- 

Ans. 263 2-dollar bills, dred 2 even hundred times, and 
and 1 dollar over. 1 hundred over ; 2 is in the 1 

hundred over, and the 2 tens, or 
in 12 tens, 60 times, or 6 ten times exactly ; 2 is in the 
7, 3 times, and 1 over. The 2 hundreds, 6 tens, and 3 put 
under 527, make 263, evidently the answer, there being 1 
remainder, which we place at the right. We proceed in 
dividing thus ; 2 is in 5, 2 times and 1 over, 2 is in 12, 6 
times, 2 is in 7, 3 times and 1 over. 


Explain how example 1, lesson 46, is performed. How do we pro¬ 
ceed in dividing 527 by 2? 




DIVISION. 


49 


The number we divide by is called the divisor, that which 
we divide, the dividend, that which we obtain by dividing, 
the quotient, and the number left, the remainder ; thus, in 
the preceding example, 2 is the divisor, 527 the dividend, 
263 the quotient, and 1 the remainder. 

2. If I buy 126 dollars’ worth of broadcloth, at 3 dollars 
a yard, how many yards do I obtain for my money ? 

Ans. 42. 

3. A gentleman gave his 2 sons 264 dollars in equal 
shares ; how much did each receive ? Ans. 132 dollars. 

4. A vessel owned by 5 persons in equal shares, earned 

1,230 dollars in one year ; what was each one’s part of 
the gain ? Ans. 246 dollars. 

5. If I sell 8 horses for 688 dollars, how much do I get 

apiece for them ? Ans. 86 dollars. 

6. How many barrels of flour can I buy for 64,459 dol¬ 
lars, at 9 dollars a barrel, and how many dollars will be 
left ? Ans. 7,162 barrels, and there will be 1 dollar left. 

7. A teamster hauled two hundred and sixteen kegs of 

lard in four equal sized loads ; how many kegs were there 
in each load ? Ans. 54. 

8. A drover wished to put 1,305 sheep in 6 equal flocks; 
how many will there be in a flock, and how many will there 
be over ? Ans. there will be 217 in a flock, and 3 over. 


Lesson 47. 


Way of dividing when the divisor consists of more than 
one figure. 

1. 13 men gained 1,963 dollars, and divided it equally ; 
what was each one’s share ? 


OPERATION. 

13)1963(1 51 dollars. Ans. 
1 3 


6 6 
6 5 


1 *3 

1 3 


TRIAL 
6 6 
52 

1 4 


Explanation. We di¬ 
vide as before, but for the 
sake of ease, write down 
all of the work thus. We 
make a curved line each 
side of 1,963, and place 
13 at the left. Now 
19 hundreds divided by 
13 give 1 hundred, which 
we place at the right of 


What is called the divisor ? Dividend ? Quotient ? Remainder ? 

In example 1, lesson 46, what is the divisor? Dividend ? Quotient: 
Remainder ? 


5 






50 


DIVISION. 


1,963, and multiplying 13 by the 1 hundred, subtract the 
product, 13 hundreds, from 19 hundreds ; we have 6 hun¬ 
dreds left, and bringing down the 6 tens we have 66 tens to 
divide by 13. Let us see if the quotient be 4 tens ; 
multiplying 13 by 4 tens we get 52 tens, which subtracted 
from 66 tens leave 14 tens. The quotient is evidently 
larger. Let us try 5 tens. Multiplying 13 by 5 tens we 
get 65 tens, which subtracted from 66 tens, leave L ten ; 
we now place the 5 tens at the right of the quotient, and 
bringing down the 3, we have 13, which divided by 13 
gives 1. This we put at the right of the other quotients, 
and multiplying 13 by it, subtract the product, which leaves 
no remainder. 

151 is plainly the whole quotient, since 13 is contained 
in 1,963, 1 hundred times, 5 ten times, and 1 time. 

2. A fisherman sold 12 shad for 108 cents ; how much 

did he get apiece for them ? Ans. 9 cents. 

3. How many times is 25 contained in 1,043, and how 
large is the remainder ? 

Ans. 41 times, and the remainder is 18. 

4. A man earns 26 dollars a month ; how long will it 

take him to earn 1,638 dollars ? Ans. 63 months. 

5. If 112 muskets are worth 1,344 dollars, what is the 

value of each ? Ans. 12 dollars. 

6. A man gave 73 dollars for 6,205 alewives ; how 

many could he have bought for 1 dollar ? Ans. 8-5. 

7. 87 cattle slaughtered at Brighton, were found to 

weigh 35,757 pounds ; if they were all of a size, what was 
the weight of each ? Ans. 411 pounds. 

8. A teacher wished to divide 412 nuts equally among 
34 boys ; how many can he give each, and how many will 
be left, that is, what will be the remainder ? 

Ans. he can give 12 to each, and there will ber4 left. 

Lesson 48. 

From what precedes we get the following 

RULE FOR DIVISION. 

Write the divisor at the left of the dividend. Then if the 
divisor consists of only one figure, draw a line beneath the 


Explain how you divide 1,063 by 13. 

Why is 151 plainly the whole quotient? 

How do you write the numbers in Division ? How, then, do you pro¬ 
ceed if the divisor consists of only one figure ? 




DIVISION. 


51 


dividend, and divide the left hand figure or two left hand 
; figures, placing the quotient beneath; the next figure of the 
dividend joined to the right of the remainder, if any, must 
now be divided, and the quotient put at the right of the other, 
and so on. If the divisor consists of two or more figures, 
find how many times it is contained in the smallest possible 
number of figures at the left of the dividend, and place the 
quotient at the right of the dividend; then multiply the divisor 
by the quotient, and subtract the product from the figures di¬ 
vided. Bring down the next figure of the dividend, join it to 
the right of the remainder, if any, and divide as before, put¬ 
ting the quotient at the right of the other, and so on. 

3 is in 7, 2 times and 1 over. 3 times 2 are 6 and 1 are 7. 

Therefore, as we sometimes make a mistake in dividing, 
we can prove the work, 

By multiplying the divisor and quotient together, and 
adding the remainder, if any, to the product; if the result be 
equal to the dividend, the work will generally be right. 

1. Divide 636 by 15. 

OPERATION. PROOF. 

1 5)636(42 quotient. 4 2 quotient. 

6 0 15 divisor. 

3 6 2 1 0 

3 0 4 2 

6 remainder. 6 3 0 

6 remainder. 

6 3 6 dividend. 

JVofe. Each example should now be proved. 

2. Divide 457,408 by 1,021. 3. Divide 10,985 by 21. 

4. Divide 3,584 by 32. 5. Divide 895 by 8. 

Lesson 49. 

1. Divide 3,647,819 by 7. 

2. Divide 46,200,981 by 3,975. 


How do you proceed if the divisor consists of two or more figures. 
How many times is 3 in 7, and how many over ? How many are 3 
times 2 added to 1 ? 

How, then, can we prove the work in Division ? 



52 


DIVISION. 


3. Divide 115,692,192 by 1,894. 

4. An owner in a privateer was to receive one dollar out 
of every 6 that she made ; after property to the amount of 
22,302 dollars had been taken by her, what was his share ? 

Ans. 3,717 dollars. 

5. Suppose the salary of the bishop of Durham, in Eng¬ 
land, to be 96,666 dollars a year, how much does he re¬ 
ceive a day, and what is the remainder, there being 365 
days in a year ? 

Ans. he receives 264 dollars a day, and the remainder is 
306 dollars. 

6. If there are 136 furlongs in 17 miles, how many fur¬ 
longs are there in 1 mile ? Ans. 8. 

7. A man has 6 chaises built in the same way ; all of 
them are worth 750 dollars ; what is the value of each ? 

Ans. 125 dollars. 

8. Divide 975 by 75. Ans. 13. 

9. Divide 5,310 by 45. Ans. 118. 

10. A trader bought 3,472 pounds of butter for 434 dol¬ 
lars; how many pounds did he get for one dollar ? Ans. 8. 

Lesson 50. 

1. A bank having 28,400 dollars in silver, the officers 

put them in 25 bags, with the same quantity in each ; how 
much was that ? Ans. 1,136 dollars. 

2. If 4 yards of cloth make one suit of clothes, how 

many suits will 108 yards make ? Ans. 27. 

3. A man bought 44 pounds of cheese for 528 cents ; 

how much did he give a pound ? Ans. 12 cents. 

4. A man who had 625 pounds of butter, put it in 25 fir¬ 

kins, with the same number of pounds in each ; what num¬ 
ber was that ? Ans. 25 pounds. 

5. What is the quotient of 225 divided by 15 ? Ans. 15. 

6. How long will 57,202 pounds of flour last, if 37 

pounds are used daily ? Ans. 1,546 days. 

7. Divide 6,665 by 1,333. Ans. 5. 

8. Divide 28,791 by 7. Ans. 4,113. 

9. If 40 bushels of potatoes are worth 1 ton of hay, how 

many tons are 364,920 bushels worth ? Ans. 9,123. 

10. Suppose I want to go 873 miles in 9 days, how far 
must I travel each day, if I ride at a uniform speed ? 

Ans. 97 miles. 


DIVISION. 


53 


Lesson 51. 

1. How many are 3 times 4 ? 3 is in 12 how many times ? 
4 is in 12 how many times ? 

Theref ore , by dividing the product of two numbers by one 
of them, we obtain the other. 

2. If you get 252 dollars for 6 loads of cider, with 14 
barrels to a load, what price a barrel are you paid ? 

OPERATION. 

14 84)252(3 dollars. Ans. 

6 2 5 2 


8 4 barrels. 

Therefore , by dividing the product of several numbers by 
the product of all of them but one , ice obtain this one . 

3. 7 men were paid 105 dollars for labor done at 1 dol¬ 
lar apiece a day; how many days did they work ? Ans. 15. 

A- A teamster hauled 65 bushels of corn at a load until 
he had carried away 1,625 bushels ; how many loads had 
he taken ? Ans. 25. 

5. 2 horses eat 4,872 pounds of hay in 3 months ; how 
much did each of them consume in a day, there being 7 
days in a week, and 4^weeks in a month ? Ans. 29 pounds. 

6. 17,160 is the product of 13, 8, 11, 3, and another 
number multiplied together ; what is that other number ? 

Ans. 5. 

7. I paid 85,120 cents for a load of butter in 95 firkins, 

giving 16 cents a pound for it ; how many pounds were 
there in a firkin ? Ans. 56 pounds. 

CONTRACTIONS IN DIVISION. 

Lesson 52. 

1. A coal merchant bought 1,133 dollars’ worth of anthra¬ 
cite coal, at 11 dollars a ton ; how many tons did he buy ? 


W hat do we obtain by dividing the product of two numbers by one 
of them ? 

What do we obtain by dividing the product of several numbers by 
the product of all of them but one ? 

5* 




54 


DIVISION. 


operation. Explanation. After bring 

1 1)1 133(103 tons. Ans. ing down the 3 tens we 

1 1 find that 11 is contained in 

•- them 0 ten times, which 

3 3 we put in the quotient ; 

3 3 and then bring down the 

-- 3, and divide 33 by 11. 

So after bringing down a figure, if the number made be 
less than the divisor, 

Put 0 in the quotient, and bring down the next figure, and 
divide . 

2. 23 horses, each of the same value, are worth 2,300 
dollars; how much is one of them worth ? Ans. 100 dollars. 

3. 42,230 bricks were hauled in 41 equal loads ; how 

many were taken in each load ? Ans. 1,030. 

4. Divide 89,760 by 17. Ans. 5,280. 

5. 27,054 acres of land were divided into 27 equal sized 
lots for sale ; how many acres were put in each lot ? 

Ans. 1,002. 

6. James had 230 chestnuts to divide among 10 children; 
how many must he give to each ? 

operation. Explanation. Since by putting one 

2 3 | 0 0 at the right of a number we multi- 

2 3 chestnuts. Ans. Pjy. it by 10, by cutting one off we 

divide by 10. See Numeration, latter 
part of lesson 5. If James had had 234 apples, we should 
have cut off one figure from the right ; thus 2 3 | 4, leav¬ 
ing the answer 23, with a remainder 4. 

7. There being 100 cents in a dollar, how many dollars 

are there in 53,000 cents ? Ans. 530. 

8. If 1,000 soldiers have 2,311 flints, how many can 
each one have, and how many will there be over ? 

Ans. each one can have 2, and there will be 311 over. 

I 

So when the divisor is 10, 100, &,c. 

Cut off as many figures from the right of the dividend as 
there arc 0s in the divisor. The figures cut off are the re¬ 
mainder, the others the quotient. 

Explain how example 1, lesson 52, is performed. 

If after bringing down a figure the number made be less than the 
divisor, what is done ? 

Explain how example 6, lesson 52, is performed. 

What if the divisor be 10, 100, &c. ? 




DIVISION. 


55 


9. Divide 599,843 by 100. 

10. 10 children inherited 15,220 dollars; what was the 
share of each ? 

11. If you divide 372,000 into 1,000 parts, how large 
will each part be ? 

12. There are 21,300 pounds of beef to be distributed 
among 10,000 men ; how many pounds can each receive, 
and how much will remain ? 

Lesson 53. 

1. A man had 330 pounds of lard which he wished to put 
into 20 equal sized pots , how many pounds must be put 
in each ? 

operation. Explanation. To multiply 

2|0)3 3|0 by 20 we multiply by 2, and 

- put the 0 at the right of the 

16 10 product ; therefore to divide 

Ans. 16 pounds, and there 330 by 20 we cut off the 0 
are 10 pounds over. from 330, and divide the rest 

by 2. The remainder 1 be¬ 
ing 1 ten, we put the 0 cut off, at the right of it. If he 
had had 337 pounds, we should have cut off the 7, and put 
it at the right of the 1 ten for a remainder. 

2. How many piles of 600 dollars can you take out of a 
heap containing 12,200, and how many will there be over ? 

Ans. 20 piles, and 200 over. 

So when there are 0s at the right of the divisor, 

Cut them off; also cut off the same number of figures from 
the right of the dividend, divide ivhai remains of the dividend 
by ivhat remains of the divisor , and place the figures cut off 
from the dividend at the right of the remainder. 

3. How many times is 4,000 contained in 2,010,000 ? 

Ans. 502 times, and 2,000 over. 

4. 450 men were paid 31,500 dollars ; what was the 

share of each ? Ans. 70 dollars. 

5. Divide 3,541 by 30. Ans. 118, and 1 over. 

6. How many divisions of 90 feet each can I make in one 
mile, or 5,280 feet ? Ans. 58, and there are 60 feet over. 

7. A party of 20 men having obtained 720 dollars, di¬ 
vided them equally ; what was each one’s share ? 

Ans. 36 dollars. 

Explain how example 1, lesson 53, is performed. 

When there are Os at the right of the divisor, how do you get the 
answer ? 




56 


PROMISCUOUS QUESTIONS. 


8. What is the quotient of 75,000 divided by 500 ? 

Ans. 150. 

9. How many acres of land, at 310 dollars an acre, can 

I buy for 1,240 dollars ? Ans. 4 acres. 

10. How long will it take a man to travel 2,800 miles, 

if he goes 70 miles a day ? Ans. 40 days. 


PROMISCUOUS QUESTIONS 

IN 

ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION. 

Lesson 54. 

To be performed in the mind. 

1. Rufus having 27 cents, wishes to buy as many oranges 

at 6 cents apiece as possible, and to spend the rest in ap¬ 
ples at 1 cent apiece ; how many oranges and apples can 
he buy ? ** 

2. If he has 25 cents, how many oranges and apples 
can he buy ? 

3. James picked up 15 apples, and William 21, but 
William soon after gave James 7 ; how many did each 
have then ? 

4. How much money do 35 10-dollar bills make ? 

5. A man having 97 sheep, purchased 35 more ; how 
many had he then ? 

6. A farmer sold 2 loads of cider ; each load contained 

4 barrels, and each barrel 30 gallons ; how many gallons 
did he sell ? 

7. A boy having 75 cents, spent 28 for a knife ; how 
many did he then have ? 

8. If a man has an income of 2 dollars a day, and spends 

5 dollars a week, how much will he save in 4 weeks ? 

9. Suppose a man on foot to be 25 miles ahead of a man 
on horseback, and to walk 5 miles an hour ; how long be¬ 
fore the man on horseback will overtake the other, if he 
rides 10 miles an hour ? 

10. A farmer having 81 dollars, bought 10 sheep at 3 
dollars apiece, and 9 sheep at 4 dollars apiece ; how much 
money did he have after paying for them ? 



PROMISCUOUS QUESTIONS. 


57 


11. If I buy a yoke of oxen, a cart, and three cows for 
240 dollars, how many 5-dollar bills must I give in 
payment ? 

Lesson 55* 

1. If you pay 4 cents for 2 oranges, what do you give 
apiece for them ? If you pay 2 cents for 4 oranges, how 
many do you get for one cent ? 

So we divide the price by the quantity to get the value of one 
thing , and the quantity by the price in cents to get what one 
cent will buy, and by the price in dollars to get what one dol¬ 
lar ivill buy. 

For the Slate. 

2. 147 tons of hay sold for 3,087 dollars ; what did it 

bring a ton ? Ans. 21 dollars. 

3. A man bought 3,684 pounds of sugar for 307 dollars; 
how many pounds did he get for one dollar ? Ans. 12. 

4. If you buy 160 acres of land at 25 dollars an acre, 

and expend 100 dollars upon it, at how much an acre must 
you sell it to gain 220 dollars ? Ans 27 dollars. 

Explanation. The whole cost added to 220 dollars, is 
what all the land must be sold for to gain 220 dollars. 

5. A merchant having 45 barrels of pork, sold 25 of 

them at 25 dollars a barrel, but the price falling he sold 
the rest at 21 dollars a barrel ; what did he get for the 
whole ? Ans. 1,045 dollars. 

6. Multiply 4,829,001 by 300,128. 

Ans. 1,449,318,412,128. 

7. A merchant bought 4,960 yards of coarse cotton cloth 
for 310 dollars ; how many yards did he get for one dollar ? 

Ans. 16. 

8. What number multiplied by 20 produces 2,740 ? 

Ans. 137 

Explanation. Remember that the divisor and quotient 
multiplied together produce the dividend. 

9. What number multiplied by 12 produces 144 ? 

Ans. 12. 

10. What number added to 192 makes 271 ? Ans. 79. 


How do we get the value of one thing from the price and quantity ? 
How do we get what one cent or one dollar will buy, from the price 
and quantity ? 




58 


PROMISCUOUS QUESTIONS. 


Lesson 56. 

1. A merchant bought 37,520 dollars worth of flour, and 

after selling it all, found he had gained 4,281 dollars ; 
what did he sell it for ? Ans. 41,801 dollars. 

2. A man bought a farm for 3,230 dollars, paying with 
a farm worth 1,240 dollars, a yoke of oxen worth 75 dol¬ 
lars, a wagon worth 83 dollars, three cows worth 54 dol¬ 
lars, and the rest in money; how much money did he pay ? 

Ans. 1,778 dollars. 

3. A barrel of pork contains 200 pounds ; now if you 
buy 3 barrels, each of which lacks 4 pounds of being full, 
at 16 cents a pound, what must you give for the whole ? 

Ans. 9,408 cents. 

How many dollars must you give, 100 cents making a 
dollar i Ans. 94 dollars and 8 cents. 

4. A merchant bought a ship for 11,200 dollars, and 

after expending 1,247 dollars in repairs, sold her for 14,000 
dollars ; what did he gain ? Ans. 1,553 dollars. 

5. 10 children inherited 4,535 dollars apiece ; what did 

they all inherit ? Ans. 45,350 dollars. 

6. If I give 141 dollars for a piece of cloth containing 47 
yards, how much must I sell it for to gain 1 dollar a yard ? 

Ans. 188 dollars. 

7. A man who owned the following quantities of land, 

500 acres, 17 acres, 121 acres, and 98 acres, sold 325 
acres ; how much did he have left ? Ans. 411 acres, 

8. What number subtracted from 2,521 leaves 178 ? 

Ans. 2,343. 

9. How long will it take a steamboat to go 345 miles, if 

shemioves 4 15 miles an hour ? Ans. 23 hours. 

10. A man sold a cargo of salt for 2,300 dollars, and 
made on it 625 dollars ; what did it cost him ? 

Ans. 1,675 dollars. 

Lesson 57. 

1. If I spend 35 dollars a month, how long will 700 

dollars last me ? Ans. 20 months. 

2. If I spend 35 dollars a month, and earn 25, how long 
before I shall spend 700 dollars more than I earn ? 

Ans. 70 months. 

3. A man has a farm worth 2,300 dollars, stock worth 


COMMON FRACTIONS. 


59 


450, and 575 in cash ; he. owes one man 1,125 dollars, and 
various others 323 dollars ; what property will he have 
when all his debts are paid ? Ans. 1,877 dollars. 

4. Divide 4,725,000 by 5,000. Ans. 945. 

5. 4 children inherited 2,250 dollars apiece, but one hav¬ 

ing died, the remaining 3 inherited the whole ; what was 
each one’s part ? Ans. 3,000 dollars. 

6. If a man rides 60 miles a day, how far will he go in 

150 days ? Ans. 9,000 miles. 

7. How long can 5 men live on 475 crackers, if they 

consume 5 apiece each day ? Ans. 19 days. 

8. Add 2,581, 37, 583, 10,000 and 12. Ans. 13,213. 

9. 12 men are hired at 1 dollar a day ; how much will 
their wages amount to in 300 days ? Ans. 3,600 dollars. 

10. Subtract 1,287 from 1,287,000. Ans. 1,285,713. 

The preceding rules, Addition, Subtraction, Multiplica¬ 
tion, and Division, comprise all the operations ever per¬ 
formed in arithmetic ; other rules are only different ways 
of applying these. 


FRACTIONS, 

OR BROKEN NUMBERS. 


COMMON FRACTIONS, 

OFTEN CALLED VULGAR FRACTIONS. 

Lesson 58. 

If you cut an apple into 2 equal parts, each part will be 
a half of the apple ; if you cut it into 3 equal parts, each 
part will be a third of the apple ; if you cut it into 4 equal 
parts, each part will be a fourth of the apple ; and if you 


What is said of the preceding rules? 

If you cut an apple into 2 equal parts, what portion of the apple 
will each part be? What portion of the apple will each part be if 
you cut it into 3 equal parts ? 4 equal parts ? 





60 


COMMON FRACTIONS. 


cut it into any other number of equal parts, the ordinal 
number , corresponding to it, is used to express the size of 
a part ; thus, if you cut it into 23 equal pieces, each one 
will be a twenty-third partif you divide it into 78 equal 
pieces, each one will be a seventy-eighth part. 

If you have a number of apples, say 6, and divide them 
into 2 equal heaps, each heap will be a half of the whole ; 
if you divide them into 3 equal heaps, each heap will be 
a third of the whole. If you have 20 apples, and divide 
them into 4 equal heaps, each heap will be a fourth of the 
whole ; if you divide them into 5 equal heaps, each heap 
will be a fifth of the whole, and so on. 

Into how many equal parts must you divide a pear so 
that each division may be a half ? how many halves are 
there in any thing then ? 

Into how many equal parts must Joseph divide a piece 
of gingerbread so that each portion may be a third of the 
whole ? How many thirds are there in any thing then ? 

Into how many equal parts must you divide any thing so 
that each part may be a fourth ? How many fourths are 
there in any thing then ? 

Into how many equal parts must you divide any thing so 
that each portion may be a fifth ? A sixth ? A seventh ? 
An eighth ? A thirteenth ? A twenty-first part ? A fortieth ? 
A hundred and fifty-sixth ? A five hundredth ? 

How many fifths are there in any thing ? Sixths ? 
Sevenths ? Ninths ? Twenty-fourths ? Forty-sevenths ? 
Seventy-sixths ? Two hundred and fifteenths ? Thou¬ 
sandths ? 

To be performed in the mind. 

1. Lucius had 2 peaches, and he gave Samuel one half 
of them ; what number was that ? How many halves of 2 
are there then ? 

2. What number is one half of 4 ? Of 8 ? Of 10 ? Of 
12 ? Of 14 ? Of 20 ? Of 30 ? Of 46 ? Of 52 ? Of 78 ? 
Of 192 ? How many halves of 4 are there then ? Of 8 ? 
Of 10 ? Of 12 ? Of 14? Of 30? Of 78 ? Of any num¬ 
ber ? 


If you cut it into any other number of equal parts, what is used to 
express the size of a part P 

What if you cut it into 23 equal pieces ? 78 equal pieces ? 

If you divide 6 apples into 2 equal heaps, what part of the whole will 
each heap be P What part will each heap be if you divide them into 3 
equal heaps ? If you divide 20 apples into 4 equal heaps ? 5 equal heaps ? 



COMMON FRACTIONS. 


61 


3. Sarah had 6 sugar plums ; after giving away all but 
one third of them, how many had she left ? How many 
thirds of 6 are there ? 

4. A man having 24 dollars, spent one fourth of it; what 
sum was that ? How many fourths of 24 are there ? 

5. What is one fifth of 45 ? One sixth of 12 ? One 
seventh of 56 ? One tenth of 100 ? 

6. How many halves are there in any number ? Thirds ? 
Fourths ? Fifths ? Sixths ? Sevenths ? Eighths ? Ninths ? 
Twelfths ? Twentieths ? Fifty-fourths ? Ninety-thirds ? 

7. Elias had 12 cents, and he gave Robert one third of 
them, and Caleb one fourth ; how many did Rooert get ? 
Caleb ? Which then is the greatest, a third or a fourth 
of 12 ? 

8. Which is the most, a half of 18 nuts or a third of 
them ? A fourth of 40 nuts or a fifth of them ? A fifth of 
60 nuts or a sixth of them ? 

9. William had an eighth of 72 cherries, and Daniel a 
ninth of them ; which had the most, and how many ? 

- 10. Which is the greatest, a half of an apple or a third? 

A third or a fourth ? A fourth or a fifth ? A fifteenth or a 
sixteenth ? A half or a fourth ? A third or a sixth ? A 
fourth or a sixth ? 


Lesson 59. 

To be performed in the mind. 

1. Mary had one half of a dollar, and her father gave 
her another half; how many halves had she then ? What 
part of a dollar had she then ? 

2. A man had one fifth of an acre of land, when he 
bought two fifths of an acre more; what part of an acre had 
he then ? 

3. Levi bought three eighths of a sheet of gingerbread 
for himself, and two eighths for his playmate ; what part 
of a sheet did he buy for both ? 

4. If you take one third of an apple from two thirds 
what part of an apple will be left ? 

5. Augustus had four fourths, or quarters, of an apple 
in his hand when he dropped one fourth ; what part of an 
apple remained in his hand ? 

6. A merchant owned six tenths of a ship, when he sold 
three tenths ; what part did he keep ? 

7. A farmer having 10 dollars, bought a number of arti- 

6 


62 


COMMON FRACTIONS. 


cles, and then counted his money ; he found two fifths of 
it left ; what sum was that ? Explanation. What is one 
fifth of 10 dollars ? Then what are two fifths ? 

8. How many fifths of his money did he spend ? 

9. Ephraim paid three fourths of 40 cents for a knife ; 
how many cents did his knife cost ? How many fourths of 
his money did he save ? 

10. Richard had 63 walnuts, but he lost two sevenths of 
them ; what number remained ? 

11. If a watermelon be worth 24 cents, how much is five 
eighths of it worth ? 

12. Oliver had two pears, each of which he cut into 
halves ; how many halves were there ? 

13. If you cut each of 3 apple-pies into fourths, how 
many fourths will there be ? 

14. How many sixths of 1 are there in 2 ? In 5 ? In 4 ? 
In 7 ? In 9 ? In 6 ? 

15. If you divide one half of an apple into 2 parts, how 
large will each part be ? How large will each part be if 
you divide one third of an apple into 2 parts ? One third 
of an apple into 3 parts ? 

Lesson 60 . 

It is usual to write fractions in figures ; thus, one half is 
written one'third one fourth one fifth a, one twen¬ 
tieth two thirds §, three fourths f-, seven tenths T 7 ^, three 
eighteenths tf- 

That is, the number shoiving the size of a part is placed 
below the line , and the number of parts is placed above the line. 

The number below the line is called the denominator , and 
the number above the line the numerator. 

How do you write in figures one ninth, one twelfth, one 
thirty-third, two sevenths, two forty-fifths, three eigh¬ 
teenths, four fifths, five elevenths, seven fourteenths, twelve 
one hundredths, twenty-five forty-firsts, ninety-two one 
hundredths, seventy-seven ninetieths, one hundred and 
twenty-four one hundred and fiftieths, three halves, five 


How is it usual to write fractions ? How is one half written ? One 
third ? One fourth P One fifth ? One twentieth ? Two thirds ? Three 
fourths ? Seven tenths ? Three eighteenths ? 

What number of a fraction do we put below the line ? Above the line* 
Which is the denominator ? Numerator ? 




COMMON FRACTIONS. 

thirds, eight fifths, seventeen twelfths, twenty-five six¬ 
teenths, two halves, three thirds ? 

How do you read f, ft, 4 , J T , |f, 1 %, £$, if, ■££,, 

«*. if. fI. H. M> #> f, A. H, W. f, f ? 

A fraction whose numerator is smaller than its demomi- 
nator, is called a proper fraction. A fraction whose nu¬ 
merator equals or exceeds its denominator, is called an improper 
fraction. 

To be performed in the mind. 

1 . If you cut 2 apples into thirds, and take of each, 
how many thirds of 1 apple will you have ? Are -f of 1 the 
same as ^ of 2 then ? 

2 . Moses cut 3 oranges into fourths, and took ^ of each; 
how many fourths of 1 orange did he get ? What part of 
1 then is \ of 3 ? What part of 3 are £ of 1 ? 

3. If you have 5 melons to divide among 6 boys, what 
part of a melon is each boy’s share ? Explanation. If you 
had but 1 melon to divide, what part would each boy have? 
How many sixth parts can each have in all the 5 melons ? 
What part of 1 is £ of 5 ? What part of 5 are f of 1 ? 

4. 10 men have equal shares in 4 large pumpkins ; how 
large a part must each have ? What part of 1 is T \j of 4 ? 
What part of 4 is ^ of 1 ? 

5. 3 boys have 7 pears ; what is each one’s share ? 
What part of 1 is ^ of 7 ? What part of 7 are £ of 1 ? 

The numerator is always supposed to be divided by the 
denominator, and the value of the fraction is the quotient. 
A quotient may always be expressed by writing the divisor 
as a denominator under the dividend ; thus % is 1 divided 
by 3,, f is 2 divided by 5. This last fraction may be con¬ 
sidered as two fifths of 1 , one fifth of 2, or 2 divided by 5. 

6 . In what different wavs may you consider and read f- ? 

if f ? W ? ? f A« ? t ? A ? H ? 

7 . How can you express the quotient of 1 divided by 2 ? 

7 divided by 3 ? 14 divided by 25 ? 10 divided by 307 ? 

120 divided by 9 ? 19 divided by 20 ? 3 divided by 47 ? 

101 divided by 1,250 ? 33 divided by 55 ? 6 divided by 50 ? 

What is called a proper fraction ? An improper fraction ? 

What is the numerator always supposed to be divided by ? What is 
the quotient? How may a quotient always be expressed ? What then 
is 5 ? § ? How may | be considered ? 



64 


COMMON FRACTIONS. 


Lesson 61 . 

To find what part of one number another number is. 

To be performed in the mind. 

1 . If there are 2 apples on the table, and 1 of them be¬ 
longs to you, what part of the 2 apples belongs to you, 

L or what part ? If you have 1 of 3 apples, what part of 
the whole are yours ? 1 of 4 ? 1 of 5 ? 1 of 6 ? 1 of 7 ? 

I of 8 ? 

2 . If you have 2 of 3 apples, what part of the whole are 
yours ? Explanation. If you have 1 of 3 apples, what 
part are yours ? Then what part are yours if you have 2 
of 3 ? 

3. What part of 4 apples are 3 apples ? What part of 5 
is 3 ? What part of 4 is 5 ? What part of 6 is 2 ? 

Therefore, to find what part of one number another num¬ 
ber is, 

Make the first the denominator of a fraction of which the 
last is the numerator . 

4. A man owns 7 acres in a piece of land that contains 

II acres ; what part of the piece does he possess ? 

5. William has 3 oranges, and Samuel 4 ; what part of 
Samuel’s number has William ? 

6 . What part of 4 is 5 ? What part of 7 is 9 ? 

7. John has 8 cents, and Jacob 5 ; what part of Jacob’s 
number has John? What part of John’s number has Jacob ? 

8 . What part of 28 is 13 ? What part of 13 is 28 ? 

9. A captain of a ship received 95 dollars, and the mate 
45 ; what part of the captain’s share is the mate’s ? 

10 . What part of 19 is 4 ? What part of 7 is 70 ? 

Lesson 62 . 

To find the exact quotient when there is a remainder . 

To be performed in the mind. 

1. 2 boys have 3 apples between them ; how many ap¬ 
ples, and what part of an apple is the share of each ? 

2 . If you divide 4 pears into 3 equal portions, how 
many pears, and what part of a pear will there be in each 
portion ? 


How do we find what part of one number another number is ? 




COMMON FRACTIONS. 


65 


3. If you divide 5 bushels of potatoes among 3 persons, 
how many bushels, and what part of a bushel will each 
receive ? How many bushels, and what part of a bushel 
will each receive if you divide 5 bushels among 4 persons ? 
6 bushels ? 7 bushels ? 9 bushels ? 10 bushels ? 

For the Slate. 

4. 9 dollars are to be paid to 4 men in equal shares ; 
what will each one have 5 

operation. Explanation. We divide 9 into 4 

4)9 parts, and get 2 dollars for each part, 

— and there is 1 dollar over. This 

2 £ dollars. Ans. divided into 4 parts gives £ of a dollar 
for each; the £ being put at the right of 
the 2 , makes 2 £ dollars for the answer. 

Therefore, when there is a remainder after dividing, 

Write the divisor beneath it; the fraction thus made, 
is a part of the quotient , and must be put at the right of it. 

Any number composed of a whole number and a frac¬ 
tion, like 2 £, is called a mixed number. 

5. A man gave 6 dollars for 5 bushels of corn ; what 

was the price a bushel ? Ans. 1 ^ dollar. 

6 . 3 boys divided 5 oranges between them ; what was 

the portion of each ? Ans. 1 §. 

7. Divide 35 by 8 . Ans. 4§. 

8 . Divide 911 by 6 . Ans. 151£. 

9. Divide 73 by 21. Ans. 3^|. 

10 . A laborer gets one bushel for every 10 that he thresh¬ 
es; what quantity will he have after threshing 27 bushels ? 

Ans. Qfjy bushels. 

11 . Divide 33 by 12. Ans. 2^. 

12 . What is the quotient of 2,700 by 185 ? Ans. 14^|^. 

Lesson 63. 

To change a whole or mixed number to an improper fraction. 
To be performed in the mind. 

1 . How many halves of a pear are there in 2 pears ? In 
2£ pears ? In 3 ? In 4 ? In 7 ? In 7£ ? In 15 ? In 20 ? 


Explain how example 4, lesson 62, is performed. 

What is done when there is a remainder after dividing? 
What is called a mixed number? 

6* 



66 


COMMON FRACTIONS. 


In 100 ? How many thirds are there in 3 pears ? In 4§ 
pears ? In 6 ? In 7 a ? In 9 ? 

2 . How many eighths of an inch are there in 3§ inches ? 
In 4f inches ? In 8 inches ? In inches ? In 12 inches ? 

8 . 2 loaves were each broken into 10 equal parts to dis¬ 
tribute among some starving sailors; how many tenth parts 
were there in both loaves ? How many tenth parts would 
there have been in 3 loaves ? In 4 ? In 6 ? In 5^ ? In 9 ? 

For the Slate. 

4 . A merchant having 13 pounds of cloves, put them in 
parcels of ^ of a pound each for sale ; how many parcels, 
or fourths of a pound did he make ? 

operation. Explanation. There are 4 fourths 

13 in 1 pound, and 13 times as many 

4 in 13 pounds. 


5 2 


of a pound. Ans. 

5 . If he had had 13f pounds, how many parcels or fourths 
of a pound would he have made ? 


OPERATION. 

1 3 f, 

4 

5 2 fourths of a pound in 13 pounds, 
add 3 fourths of a pound. 

5 5 


Explanation. 
There are 4 fourths 
in 1 pound, and 13 
times as many in 
13 pounds. Add¬ 
ing the 3 fourths 
to the 52 fourths 
we get 55 fourths. 


of a pound. Ans. 


Therefore, to change a whole or mixed number to an 
improper fraction, 

Multiply the whole number by the denominator , add the 
numerator to the product , if it be a mixed number, and place 
the denominator beneath the result. 


6 . Change 4^ to an improper fraction. Ans. 

7. Change 873^ to an improper fraction. Ans. £4^. 

8 . Change 35 to an improper fraction, of which the de¬ 
nominator shall be 3. Ans. -Mp. 


Explain how example 4, lesson 63, is performed. 

Explain how example 5, lesson 63, is performed. 

How do we change a whole or mixed number to an improper fraction ? 



COMMON FRACTIONS. 


67 


9. Joshua has 12f dollars in ninepences or eighths; how 

many eighths of a dollar has he ? Ans. 

10. A number of men paid ^ of a dollar apiece for ad¬ 
mittance to see a juggler ; he obtained 15 T 6 F dollars ; how 
many sixteenths of a dollar were there in that sum ? 

Ans. 

11. In 43 dollars how many sixths of a dollar are there ? 

Ans. 2 ^ s -. 

12. A farmer divided 4 bushels, 5 bushels, and § of a 
bushel of corn among some poor persons, giving each £ of 
a bushel; how many eighths of a bushel did he give away ? 

Ans. 


Lesson 64. 

To change an improper fraction to a whole or mixed number. 
To be performed in the mind. 

1. How many apples are there in § of an apple ? In § 
of an apple ? In ? In % ? In $ ? In § ? In § ? In % ? 
In ? In ? In ^ ? In ? In 4^ ? In ? 

2. A man cut up some pies into fourths to sell; after 
selling a number of pieces, or fourths, he found he had 
left ; how many whole pies would these pieces make ? 

3. A New England ninepence is ^ of a dollar ; now how 
many dollars are there in 8 ninepences, or in f- of a dollar ? 
In f of a dollar ? In ? In ? In ? 

For the Slate. • „ 

4. A merchant had of a pound of cloves ; how many 
pounds had he ? 

operation. Explanation. J of a pound make 

4)52 1 pound ; then, evidently, make 

- as many pounds as 4 goes times in 52. 

1 3 pounds. Ans. 

5. If the merchant had had of a pound, how many 
pounds would that quantity have been ? 

operation. Explanation. 4 is in 55, 13£ 

4)55 times. 

1 3 j pounds. Ans. 


Explain how examples 4 and 5, lesson 64, are performed. 



68 


COMMON FRACTIONS. 


Therefore, to change an improper fraction to a whole 0 * 
mixed number, 

Divide the numerator by the denominator , and the quotient 
will be the whole or mixed number. 

1 

6. William had G f a dollar ; how many dollars did 

he have ? A ns. 48. 

7. Some sailors, cast away on an island, lived ff- of a 
day without food ; how many days was that time ? 

, Ans. l£f- day. 

8. A grocer has 27 half-barrels of flour, that is, 23. of a 
barrel ; how many barrels are there in this quantity ? 

Ans. 134* 

9. What whole number is equal to ? Ans. 821, 

10. What whole number is equal to ? Ans. 900. 

11. Change -fff- 1 - to a mixed number. Ans. ASOfffg. 

12. Change to a mixed number. Ans. 11 T 5 ^* 

Lesson 65. 

To change a fraction to a simpler form. 

If We multiply the denominator and numerator of a frac¬ 
tion, say of by 2, for instance, it becomes f , and is 
plainly of the same value as ^ ; since 3 times 1 third make 
f, or 1, and 3 times 2 sixths make f, or 1. Now if we 
divide both denominator and numerator of f by 2, we get 
£ again, of the same value as f y but in a simpler form. 

So, if the denominator and rtumerator of a fraction be both 
multiplied , or both divided by the same number , the value of 
the fraction will not be altered ; but if both be divided by the 
same number, without a remainder , the new f raction is simpler. 

Therefore, to change a fraction to a simpler form, 

Divide the denominator and numerator by any number that 
is contained in both without a remainder. 


How do we change an improper fraction to a whole or mixed 
number ? 

If we multiply the denominator and numerator of J by 2, what frac¬ 
tion does it become ? What is its value ? Why ? Now if we divide both 
denominator and numerator of | by 2, what fraction do we get ? What 
is said of its value and form ? 

What if the denominator and numerator of a fraction be both multi¬ 
plied, or both divided by the same number ? What if both be divided 
by the same number, without a remainder ? 

How do we change a fraction to a simpler form ? 



COMMON FRACTIONS. 


69 


To be performed in the mind. 

1. If you have £ of an apple, in what simpler form can 
you express that part ? 

2. In what simpler form can you express £ ? ? 

3. Henry has £§ of a dollar ; in what simpler form can 
you express this amount of money ? 

4. A boy has of a melon ; how much is that, ex¬ 
pressed in a simpler form ? 

5. In what simpler form can you express fu ? 

6. In what simpler form can you express ? £$# ? 

7. In what simpler form can you express ? 

8. In what simpler form can you express ? 

9. In what simpler form can you express ? 

10. In what simpler form can you express -f£ ? # ? & ? 

»? 

We frequently meet with fractions composed of a large 
number of figures, rendering calculations tedious, but 
which can be changed to very simple forms ; thus £££ is 
equal to £, because if we divide both denominator and nu¬ 
merator by 124 we get 

A fraction is in its simplest form when no number but 1 
will divide both denominator and numerator without a 
remainder. 

The greatest number that will divide both denominator 
and numerator, without a remainder, is called the greatest 
common divisor; and after dividing by such a number, the 
fraction is evidently changed to its simplest form. 

Lesson 66. 

To find the greatest common divisor , and change a fraction 
to its simplest form. 

For the Slate. 

1. A ship was divided into 42 shares, and sold to various 
persons ; one bought 12 shares, and of course then owned 
£§. of the ship ; what is the simplest form to express what 
part of the ship he owned ? 


In what simpler form can we write §ff ? Why ? 

When is a fraction in its simplest form ? 

What is called the greatest common divisor of the denominator Mid 
numerator of a fraction, and when is the fraction changed to its simplest 
form ? 





70 


COMMON FRACTIONS. 


Explanation. 12 goes in 
42, 3 times, and 6 remains. 
6 goes in 12, 2 times, with 
no remainder. 6 will there¬ 
fore divide 3 times 12, or 
36, without a remainder, 
and also 36 and 6, or 42 ; 
moreover it is the greatest 
common divisor of 12 and 
42 ; for the greatest com¬ 
mon divisor of 12 and 42 
will divide 3 times 12, or 36, and 42, without a remain¬ 
der, and it must.be contained at least once more in 42 
than in 36, and no number greater than their difference, 6, 
will go in 42 one time more than in 36. 

Therefore, to find the greatest common divisor of any 
two numbers, say of the denominator and numerator of a 
fraction, 

Divide the larger number hy the smaller , ajid then the 
divisor hy the remainder , if there he any, and so on, always 
dividing the last divisor by the last remainder until nothing 
is left , when the last divisor will be the greatest common 
divisor. 

Note. The denominator and numerator of many fractions have no 
common divisor except 1. 

2. If a man buys 12 shares in a ship in which there are 

44 shares, and therefore owns \\ of her, what is the sim¬ 
plest form to write his part in ? Ans. X 3 T . 

3. Change to its simplest form. Ans. §§. 

4. Change *j-f- to its simplest form as a fraction ? Ans. 

5. If I am building 35 rods of stone wall, after I have 

finished 14 rods, what part of the whole, expressed in its 
simplest form, have I built ? Ans. §. 

6. What is the simplest form in which to express 

l,000||f? ^ Ans. l,000f§, 

7. Change to its simplest form as a fraction ? 

Ans. 

8. A man owns 10 out of 100 shares in a^actory ; what 
is his part of the property expressed in the simplest way ? 

Ans. -jAy. 

Explain how example 1, lesson 66, is performed. 

How do we find the greatest common divisor of any two numbers? 

What is said of the denominator and numerator of many fractions? 


OPERATION. 

1 2)42(3 
3 6 

6 ) 12(2 

1 2 

6 greatest common divisor. 
12 divided by 6 gives 2 
42 divided by 6 gives 7 




COMMON FRACTIONS. 


71 


9. Change T 5 C ^- to its simplest form. Ans. 

10. Change to its simplest form. Ans. x^f D . 

11. Change to its simplest form as a fraction. 

Ans. 

12. Change to its simplest form. Ans. 


Lesson 67. 

To change fractions to a common denominator. 

If we have to add to §, we obviously add the numera¬ 
tors, and get f or 1. In this case the fractions have a 
common denominator 3 ; but we cannot so easily add £, 
§, and §, where the denominators are different. In cases 
like this, before adding, we must contrive to make the de¬ 
nominators alike without altering the value of the fractions. 


For the Slate. 


1. Change £, §, and § to a common denominator. 

OPPR ATIOIV first second 

i . numerator. numerator. 


2 first denominator. 

3 second denominator. 


third 

numerator. 

2 

2 


6 

5 third denominator. 


3 4 4 

5 5 3 


3 0 common denominator. 1 5 2 0 1 2 

Ans. §£ and £§• 

Explanation. By examining the preceding operation, 
we find each denominator is multiplied with the others for 
a common denominator, and each numerator with the same 
numbers as its denominator for a new numerator ; so that 
the values of the fractions are not altered. See first part 
of lesson 65. 


Therefore, to change fractions to a common denominator, 

Multiply all the denominators together for a common de¬ 
nominator , and each numerator by all the denominators , ex¬ 
cept its own^for a new numerator. 


How do we add £ and § ? 

Where the denominators are different, as in §, g, and §, what must we 
contrive to do before adding? 

Explain how you change §, and § to a common denominator. 

How then do we change fractions to a common'denominator ? 



72 


COMMON FRACTIONS. 


2. Change f and ^ to a common denominator. 

Ans. ff, and £§. 

3. Change §, £, and § to a common denominator. 

Ans. -tsu, an( ^ 

4. Change 2£ and f to a common denominator. 

Ans. £J, and §£. 

Explanation . Change 2£ to an improper fraction first. 

5. Change 3^ and to a common denominator. 

Ans. -f-f, and . 

6. Change f, £, and ^ to a common denominator. 

Ans. If, ££, and ££. 

7. Change 8 and f to a common denominator. 

Ans. and f. 

8. Change 7£, 5£, and 1^- to a common denominator. 

Ans. -yjjft, - 1 //, and f£ 

9. Change and y 8 ^ to a common denominator. 

Ans. -j 2 5 (nnj> and 

10. Change £, and £ to a common denominator. 

Ans. £, §, and 

Explanation. We change to a common denominator by 
merely changing £ and £ to eighths. We can often abridge 
the rule in this way, by multiplying the denominators and 
numerators by such numbers as will make the denominators 
alike. 

11. Change , ££$, and ^ to a common denominator. 

12. Change and to a common denominator. 

13. £, £, y 2 ^, and ^ to a common denominator. 

ADDITION OF FRACTIONS. 

Lesson 68. 

To be performed in the mind . 

1. A little boy had T \j of a dollar, when his father gave 
him fSj more , what part of a dollar had he then ? What 
is the simplest form to express the answer in ? 

2. A man having f of a barrel of mackerel at home, 
bought § of a barrel; what quantity had he then ? If he 
had purchased 2^ barrels, what quantity would he have 
had with that at home ? 

3. What is the sum of £, f, and f- ? What is the sum 
of £, f, and 3£ ? 

4. Jane has £ of a sheet of paper, £ of a sheet, and 2 
sheets ; how much paper has she ? 


COMMON FRACTIONS. 


73 


For the Slate. 

5. A farmer sold £ of a ton of hay at one time, and § 
of a ton at another time ; what was the whole quantity he 
sold ? 

OPERATION. 

4 3 2 

3 3 4 

1 2 common 9 8 

denominator. A t-2 

■fir and A make ££ of a ton ; or, 
changing to a mixed number, 1 T ^ 
ton. Ans. 

Therefore, to add fractions, 

Add the numerators , taking care to change the fractions 
to a common denominator first, if the denominators he not 
alike. 

6. If I sell from a piece of cloth, at different times, of 

a yard, f of a yard, and | of a yard, what is the whole 
quantity I dispose of? Ans. 1£§£ yard. 

7. A farmer sold 2 bushels of potatoes to one person, 2£ 

to another, and | of a bushel to a third ; what was the 
whole quantity he sold ? Ans. bushels. 

Explanation. What number do £ and £ make ? What 
number then do 2, 2, and 1 T ^ make ? 

8. Add £, f, £, and Ans. 2£$, the fraction being ex¬ 
pressed in its simplest form. 

9. John has 5£ dollars, £, and T \j- of a dollar ; what 

is the whole amount ? Ans. 5££. 

10. A man bought of a grocer 12£ pounds of sugar, at 

one time, and 9£ at another ; what was the whole number 
of pounds he bought ? Ans. 22£. 

11. Add ^j, ££, and together ? Ans. 3^. 

12. A man owns ^ of the property belonging to a bank, 

one of his sons owns ^ of it, and another £ of it ; what 
part of it do they all own ? Ans. 


Explanation. We 
first change £ and § to 
a common denominator, 
and then £ becomes 
and §, A > adding 9 
twelfths and 8 twelfths, 
we get of a ton ; or 
changing to a mixed 
number, 1^ ton. 


Explain how example 5, lesson 68, is performed. 
How do we add fractions? 

7 



76 


COMMON FRACTIONS. 


Therefore, to multiply a whole number and a fraction. 

Divide the denominator by the whole number when it can be 
done without a remainder , and when not , multiply the numera¬ 
tor by the whole number. 

6. A farmer agreed to give a laborer of all the pota¬ 

toes he dug ; after digging 48 bushels what quantity must 
he receive ? Ans. 9 bushels. 

7. 5 men owned of a tract of land apiece ; what part 

of it do they all own ? Ans. f. 

8. What are 25 times 2|~§ ? Ans. 69. 

Explanation. What are 25 times 2 and 25 times || ? 

9. 8 men possess 2,400 dollars in equal shares ; what is 
each one’s part, or | of the whole ? Ans. 300 dollars. 

10. Multiply f by 30. Ans. 24. 

11. Multiply 45 by Ans. ££• 

12. 16,000 acres of land were divided into lots, each of 

which contained T f^ of the whole ; how much land was 
there in each lot ? Ans. 320 acres. 


Lesson 71. 

To multiply a fraction by a fraction. 

To be performed in the mind. 

1. If you cut an apple into 3 equal portions, what part 
of an apple will each portion be ? If you now cut each of 
these portions into halves, what part of an apple will each 
of the new portions be ? What is ± of ^ then ? What is ^ 

-k ^ i • 

2. What is £ of £ ? £ of -J ? 

3. What part of an orange is £ of £ ? of -£■ ? § of £ ? 

I of f? 

4. A trader sold § of 3 half-barrels of flour, that is, § of 
£ of a barrel; what quantity was that ? 

For the Slate. 

5. A man having § of an acre of land, sold f of it ; how 
much land did he sell ? 


How do we multiply a whole number and a fraction ? 



COMMON FRACTIONS. 


77 


3 

4 

12 


Explanation. 
We first multi¬ 
ply the denom¬ 
inator 3, of f, 
by the denom¬ 
inator 4, of f, 
which is the 
same as taking f of the 2 thirds ; because the 2 thirds are 
then divided by a number 4 times as large as before, and 
of course the quotient is £ of what it was. Having now £ 
of §, we multiply the numerator 2, by 3, which gives 3 
times f of §, or f of §. 


OPERATION. 

denominator of § 2 numerator of § 
denominator of f 3 numerator of f 

6 

fa or £ of an acre. Ans. 


Therefore, to multiply one fraction by another, 

Multiply the denominators together for a new denominator , 
and the numerators together for a new numerator. 


6. A man who owned fa of a ship, bequeathed to his 
son | of his portion; what part of the ship was that ? Ans. f. 

7. If you have 3£ dollars, and spend £ of it, what sum 

will that be ? Ans. If dollar. 

Explanation. Change 3£ to an improper fraction first. 

8. Multiply 2f by If. Ans. 3-f^. 

9. Multiply f of f by f. Ans. -j^. 

10. What is £ of f of f ? Ans. fa. 

11. A boy having fa of a dollar, spent f of it ; what 

part of a dollar was that ? Ans. ff . 

12. A manufacturing company divided 2dollars on 
each share owned in it ; what was paid on 7£ shares ? 

Ans. 189f dollars. 


DIVISION OF FRACTIONS. 

Lesson 72. 

To divide a fraction by a whole number. 

To be performed in the mind. 

1. If 3 pecks of corn are worth f of a dollar, what is 
one peck worth ? 

2. If 2 apples are worth ^ of a cent, what is one apple 
worth ? 


Explain how example 5, lesson 71, is performed. 
How do we multiply one fraction by another ? 
7* 



78 


COMMON FRACTIONS. 


3. 2 boys have f of a piece of gingerbread ; what is the 
share of each ? What would be the share of each if they 
had | of a piece ? § ? f ? f ? | ? 

4. f of a bushel of oats was given to 4 horses ; what 
part of a bushel did one horse have ? 


For the Slate. 

5. 2 men owned equal shares of f of a ship ; what part 
of the ship belonged to each ? 

Explanation. We divide 8 ninths into 2 
equal parts, and get f for the answer. We 
can also get the answer by multiplying the 
denominator 9, by 2 ; for if we take the 
answer just obtained, f, and multiply both 
4 and 9 by 2, which will not alter the value 
of the fraction, we shall evidently get the 
numerator of f, and the denominator mul¬ 
tiplied by 2. 


OPERATION. 

2)8 

4 

f. Ans. 

ALSO. 

9 
2 

1 8 

ft or f. Ans. 

Therefore, to divide a fraction by a whole number, 

Divide the numerator when it can he done without a re¬ 
mainder, and when not, multiply the denominator. 


6. 10 boys share £ of a dollar equally ; what is the part 

of each ? Ans. of a dollar. 

7. Divide of by 4. Ans. ^§. 

Explanation. First change 5f to an improper fraction. 

8. Divide if by 7. Ans. 

9. is 17 times what number ? Ans. fff. 

10. A man owning ff of a factory, divided his part into 

5 portions for sale ; what part of the factory did he put in 
each portion ? Ans. 

11. If 25 men are to receive lOOf dollars in equal shares, 

what will be the share of each ? Ans. 4^ dollars 

12. Divide by 105. Ans. T ff a . 

13 of an acre produced 25 bushels of corn ; what 

part of an acre produced one bushel ? Ans. 


Explain how example 5, lesson 72, is performed. 
How do we divide a fraction by a whole number ? 




COMMON FRACTIONS. 


79 


Lesson 73. 


To divide a whole number by a fraction. 


To be performed in the mind. 

1. How many times is £ of an apple in 5 apples ? 

2 How many times must a man carry away f of a 
bushel of potatoes to get 12 bushels ? Explanation. How 
many times must he carry away 1 third of a bushel to get 12 
bushels ? Then how many times must he carry away 2 
thirds of a bushel to get 12 bushels ? 

3. How many yards of cloth can I buy for 27 dollars at 
f- of a dollar a yard ? 

4. How many times is § contained in 20 ? 4 in 20 ? 

| in 12 ? 


For the Slate. 

5. How many perches of stone can I buy for 16 dollars, 
at £ of a dollar a perch ? 

Explanation. If the price had 
been £ of a dollar, I could evi¬ 
dently have bought 8 times 16, or 
128 perches, but the price being 
5 times as much, that is £ of a 
dollar, I must divide 128 by 5 to 
get the answer. 

Therefore, to divide a whole number by a fraction, 
Multiply by the denominator , and divide by the numerator. 


OPERATION. 

1 6 
8 

1 2 8 

-^f-* or 25f perches. Ans. 


6. A number of men are to be paid £ of a dollar apiece; 

how many will 56 dollars pay ? Ans. 64. 

7. How many yards of broadcloth can I buy for 110 

dollars at 5£ dollars a yard ? Ans. 20. 

Explanation. Change 5£ to an improper fraction first. 

8. Divide 87 by Ans. 36^. 

9. 100 is ^ of what number ? Ans. 1,200. 

10. Divide 112 by §. Ans. 298§. 

1 1. Divide 42 by T 8 g^. Ans. 80. 

12. How many barrels of flour can I buy for 62 dollars 

at 7f- dollars a barrel ? Ans. 8. 

13. How far can I travel for 25 dollars at ^ of a dollar 

a mile ? Ans. 400 miles. 


Explain how example 5, lesson 73, is performed. 
How do we divide a whole number by a fraction ? 




80 


COMMON FRACTIONS. 


Lesson 74. 


OPERATION. 

4 3 

3 8 

12 2 4 

or 2 penknives. Ans. 


To divide a fraction by a fraction. 

To be performed in the mind. 

1. If you can get a pound of butter for £ of a dollar, how 
many pounds can you get for £ of a dollar ? 

2. How many gallons of molasses ean I obtain for | of a 
dollar, the price of one gallon being § of a dollar ? 

3. If you give f of a dollar for a sheet of drawing-paper, 
how many sheets of such paper can you get for f of a 
dollar ? For £ of a dollar ? For f of a dollar ? 

4. How many times is § contained in § ? 

For the Slate. 

5. How many penknives at § of a dollar apiece can I 
buy for £ of a dollar ? 

Explanation. We divide £ by 
3, the numerator of f, by multi¬ 
plying the denominator 4 by 3 ; 
as we now have divided £ by a 
number 8 times too large, we 
multiply the numerator by 8. 

Therefore, to divide a fraction by a fraction, 

Divide by the numerator of the divisor , and multiply by its 
denominator. 

Note. Divide and multiply as directed in lessons 72 and 70. 

6. A man wishes to give some boys of a dollar apiece; 

how many can he give that sum to, if he has T 9 <y of a dol¬ 
lar ? ' Ans. 6. 

7. Divide 6f by T ^. Ans. 22§. 

Explanation. Change to an improper fraction first. 

8. Divide 10-/^ by 1£. Ans. 8f. 

9. Divide 2f by Ilf. Ans. ¥ 9 T 6 T . 

10. How many yards of cloth at f of a dollar a yard, 

can I get for £ of a dollar ? Ans. 6§. 

11. A man who owned of a ship, said that his part 
was ten times as large as that of another man, who owned 

; was he correct ? Ans. No. How many times as large 
was his part ? Ans. 8 times. 

12. Divide by f. Ans. f. 

13. Divide ^ by T § 0 . Ans. 5 -fc 


Explain how example 5, lesson 74, is performed. 
How do we divide a fraction by a, fraction ? 



DECIMAL FRACTIONS. 


81 


DECIMAL FRACTIONS. 


Lesson 75. 


Common fractions are sufficient for all arithmetical op¬ 
erations ; but there is still another sort, extremely con¬ 
venient and useful. We have seen in Numeration, that 
in whole numbers, a figure put at the right of another is 
ten times as small, or only one tenth as large as though it 
was in the place of that other figure. This regular dim¬ 
inution in value, is continued beyond units, so that the first 
figure at the right of units is one tenth part as much as 
though it was in the units’ place, the second figure one 
hundredth part as much as though it was in the units’ 
place, the third figure one thousandth part as much as 
though it was in the units’ place, and so on. Such figures 
at the right of units are called decimal fractions. 

A point is always placed at the left of decimal fractions 
to distinguish them from whole numbers ; as in the follow¬ 
ing table. 


3 

a 

® GO 

3 -o 

2 a 

cd 


3 

° o M* 

*• « f O ® 

£ -o z, -a 

o £ ^ £ £ 

3 ^ ^ 


m g 

•B « 

* « g 3 •s S 

S °£ 3 « 3£: 0 
® ® K ® J2 5J 


a c 
M 3 


Eh a 


TO 

. M 

2 3 

S .2 

11 

3 I 
3 3 

£ Eh 


4 8, 52 6, 45 9. 360287 1 


On the preceding principle we find, 

25.3 to be 25^. 

25.37 to be 25^ and t £q- or 25^^. 

25.371 to be 25-j^y and and yuVg- or 25-^j^y. 

25.3714 to be 25-j^y and y§"jj and 

i o tv u or 25 -j^^. 


For what are common fractions sufficient ? What have we seen in 
Numeration? How is this regular diminution in value continued? 
What are called decimal fractions? 

What is always placed at the left of decimal fractions? For what 
purpose ? 



82 


DECIMAL FRACTIONS. 


It appears, then, that we get the value of a decimal frac¬ 
tion in a common fraction, by writing a denominator be¬ 
neath, consisting of 1, with as many Os at the right of it as 
there are figures in the decimal fraction. A decimal frac¬ 
tion is always read as though it was written in such a com¬ 
mon fraction. 


Read the following numbers from your slate. 


1. ... 

.. .05 

6. 

.. 458.00037 

11. ... 

2. ... 

. 3 5 

7. 

.4847213 

12. ... 

3. ... 

.. 205 

8. 

.. 375.802 

13. ... 

4. ... 

47.537 

9. 

_ 6.2801 

14. ... 

5. ... 


10. 


15. ... 


.. .12222 
. 3.333 
.. .000001 
19.30003 
50.008 


When there are no tenths, no hundredths, or no thou¬ 
sandths, &c. in a decimal fraction, 0 is put in place of 
tenths, hundredths, or thousandths, &c. ; thus is writ¬ 
ten .02 in decimals, yx&ir is written .002, and is 

written 6.0102. 

0 prefixed to a decimal fraction, that is, placed before it, 
makes it ten times as small as before ; thus .5 is T ^, .05 is 
t&o, and .005 is 0 annexed to a decimal fraction, 

that is, placed after it, does not alter its value ; thus .5 is 
^y, .50 is equal to and .500 is equal to -j^>. 

Write the following numbers in figures on your slate. 

1. Five, and six tenths. 

2. Twelve one hundredths. 

3. Four one thousandths. 

4. 700, and 375 one thousandths. 

5. Six ten thousandths. 

6. Two hundred, and one one hundred thousandth. 

7. Twenty-five, and forty-seven thousand and twelve ten 
millionths. 

8. Seventeen hundredths. 

9. Fifty-six thousandths. 

10. One hundred and one thousandths. 


How do we get the value of a decimal fraction in a common fraction ? 
How is a decimal fraction always read? 

When there are no tenths, no hundredths, or no thousandths, &c., 
what is put in the place of tenths, hundredths, or thousandths? How 

then is 2 written in decimals ? _ 2 ? 6 102 ? 

TOO loOC TooOo 

What is the effect if 0 be prefixed to a decimal fraction, that is, placed 
before it? What then is .5 ? .05 ? .005 ? What is the effect if 0 be an¬ 
nexed to a decimal fraction, that is, placed after it ? What then is .5 ? 
.50? .500? 















DECIMAL FRACTIONS. 


83 


11. 12.000, and 12 ten thousandths. 

12. 1.000.000, and 505 ten thousandths. 

13. Two hundred and five, and six one hundred thou¬ 
sandths. 

14. One thousand one hundred and eighteen ten thou¬ 
sandths. 

15. 16, and 12 hundredths. 

16. Eleven billionths. 

l 

Lesson 76. 


To change a common fraction to a decimal fraction. 


OPERATION. 

2) 1.0 ( .5. Ans. 
1 0 


For the Slate. 

1. 2 boys are to have equal shares in 1 apple ; what 
will be each one's part expressed in decimals ? Observa¬ 
tion. It is plain that the share of each in common frac¬ 
tions will be 4 ; we obtain the value of £ in decimals as 
follows, 

Explanation. We cannot divide the 
numerator 1, by 2, the denominator, so 
we annex 0 to 1, and consider it as 10 

- tenths ; 10 tenths divided into 2 parts 

give 5 tenths, or .5 for each part. 

2. Change ^-to a decimal fraction. 

operation. Explanation. We cannot 

2 7) 2.0 0 ( .0 7 407. Ans. divide 2 by 27, so we annex 

1 8 9 0 to 2, and consider it as 20 

•- tenths; but we cannot divide 

1 1 0 20 tenths by 27, so we an- 

10 8 nex another 0 to 2, and con- 

- sider the dividend as 200 

hundredths, which divided 
into 27 parts gives 7 hun- 

- dredths for each part with 11 

1 1 hundredths remainder. The 

11 hundredths remainder we 
change to thousandths by annexing 0 ; we then divide by 
27, and get 4 thousandths with 2 thousandths remainder, 
and so on. There are no tenths in the quotient, the first 


200 
1 8 9 


Explain how example 1, lesson 76, is performed. 
Explain how example 2, lesson 76, is performed. 






84 


DECIMAL FRACTIONS. 


figure being 7 hundredths, we therefore put 0 in the tenth’s 
place. We evidently get as many decimals in the quotient 
as we annex Os to the dividend and remainders. 

Note. The operation can be carried further than has been done here, 
but the next figure obtained, being 1555555 of 1 , is too small to be no¬ 
ticed ; indeed it is not generally necessary to carry the operation so far 
as we have done, for practical purposes. 

Therefore, to change a common fraction to a decimal 
fraction, 

Divide the numerator by the denominator , annexing as 
many Os to the numerator , and to each remainder , as may be 
necessary in order to perform the division , and to carry the 
operation as far as desired. Place your point in the quotient 
so as to cut off as many figures for decimals as there have 
been Os annexed to the numerator and remainders. 

If there be not figures enough in the quotient for deci¬ 
mals, supply the deficiency by prefixing Os. 

3. 4 men had equal shares in 3 dollars ; what was the 

portion of each in decimals ? Ans. .75 of a dollar. 

4. Change § to decimals. Ans. 1.5. 

5. Change to decimals. Ans. .015625. 

6. 12 men owned 45 bushels of corn in equal shares ; 
how many bushels and parts, in decimals, did each own ? 

Ans. 3.75 bushels. 

7. Change 2 -^yir to decimals. Ans. .00125. 

8. 7 acres of land were divided into 5 equal parts ; 
what portion of an acre was each part in decimals ? 

Ans. 1.4 acre. 

9. Change | to decimals. Ans. .625. 

10. Change 2J-f to decimals. Ans. 2.56. 

Lesson 77. 

To write decimals when some are omitted. 

1. If 30 dollars are to be paid to 7 men in equal shares, 
how many dollars and cents, or hundredths of a dollar, 
should each receive ? 


How many decimals do we get ? 

How do we change a common fraction to a decimal fraction? 
What if there be not figures enough in the quotient for decimals ? 



DECIMAL FRACTIONS. 


85 


operation. Explanation. As we are to 

7 ) 3 0 express nothing less than hun- 

--- dredths, we write 4.29, and not 

4. 2 8 5 7 1 &c. 4.28, for 4.29 is nearer 4.28571, 

4.29 dollars, or 4 dol- &,c. than 4.28, since the value of 
lars and 29 cents. Ans. the first decimal omitted is .005, 
or half of .01, and of course the 
value of all the decimals omitted, or .00571, &c., is more 
than half of .01. 

If we express but two decimals in the numbers 4.286, 
4.287, 4.288, or 4.289, it is also plain that we should write 
4.29 and not 4.28. 

If we express but two decimals in the numbers 4.2847, 
4.2837, 4.2827, or 4.2817, we must write 4.28 and not 4.29, 
because the value of the decimals omitted is less than .005, 
or half of .01. 

Therefore, when we omit any decimals, 

Add 1 to the last decimal expressed , if the first decimal 
omitted be more than 5, or if there be more than one decimal 
omitted , and the first be 5. 

If the first decimal omitted be less than 5, the last deci¬ 
mal expressed should not be increased by 1. 

2. Change § to decimals. Ans. .6667, nearly. 

3. Change to decimals. Ans. .0833, about. 

4. 13 acres of land are to be divided into 7 equal parts ; 

how much in acres and decimals will there be in each 
part ? Ans. 1.857 acres, about. 

5. Change to decimals. Ans. .070885, nearly. 

6. A man gave 74 dollars for 311 bushels of potatoes ; 
how many bushels did )ie get for 1 dollar ? Ans. 4.2, about. 

To change a decimal fraction to a common fraction. 

To change a decimal fraction to a common fraction, 
proceed as directed in lesson 75, that is, 


Explain how example 1, lesson 77, is performed. 

Mention some other instances in which the last decimal expressed is 
to be increased by 1. 

Mention some instances in which the last decimal expressed is not to 
be increased. Why not? 

When should we add 1 to the last decimal expressed ? When not ? 

8 




86 


DECIMAL FRACTIONS. 


Write a denominator beneath consisting of 1 with as many 
Os annexed as there are decimals. 

The common fraction thus obtained can be changed to 
its simplest form if desired. 

7. What is the simplest form in which .75 can be ex¬ 
pressed in a common fraction ? Ans. f-. 

8. Change .00125 to a common fraction, and this frac¬ 
tion to its simplest form. Ans. -g-^y. 

9. What is the simplest form in which six hundred and 

twenty-five one thousandths can be expressed as a common 
fraction ? Ans. ■§-. 

10. Express 35.36 in a whole number and a common 

fraction in its simplest form. Ans. 35^-. 

11. What common fraction in its simplest form is equal 

to .688 ? Ans. T *y§_. 


ADDITION OF DECIMALS. 


Lesson 78. 


1. A farmer sold at different times the following quanti¬ 
ties of hay, 14.125 tons, .75 of a ton, .5 of a ton, 1.0625 
ton, and 4 tons ,* what was the whole quantity he sold ? 
operation. Explanation. We write the quan¬ 

tities with tenths under tenths, 
hundredths under hundredths, &c., 
and as ten one thousandths make 
one hundredth, ten one hundredths 
make one tenth, and ten tenths 
make one, it is plain that we must 
add and carry just as in whole 
numbers. 


1 4.1 25 

.7 5 
.5 

1 .0 6 2 5 
4. 

2 0.4 3 7 5 tons. Ans. 


Therefore, to add when there are decimals, 

Write the numbers with tenths under tenths , hundredths 
under hundredths, Spc., and then add as in ichole numbers; 
being careful to put a point at the left of the tenths in all cases. 

2. Add .00004, .27, 451, and 13.003. 

3. Rufus had f of a dollar, Josiah § of a dollar, Elias f- 

How do we change a decimal fraction to a common fraction? 

What can be done to the common fraction thus obtained ? 

Explain how example 1, lesson 78, is performed. 

How do we add when there are decimals ? 




DECIMAL FRACTIONS. 


87 


of a dollar, and Philip a. of a dollar; change each of these 
fractions to decimals, add them, and tell what they all had. 

Ans. 2.717 dollars, nearly. 

4. Change the following fractions to decimals, and add 
them, and the whole numbers ; 7 T 9 Ty , 

Ans. 12.1944, about. 

5. A man sold .875 of a cord of wood to one person, 

2.0625 cords to another, and .25 of a cord to a third ; how 
much did he sell to all of them ? Ans. 3.1875 cords. 

6. Add 27.01, 251, .3801, and 3.8. Ans. 282.1901. 

7. What is the sum of .75, .25, and .50 expressed in 

whole numbers and common fractions ? Ans. 1£. 

8. What is the sum of five, and seventy-five hundredths, 

four, and eight thousandths, and two hundred and three 
ten thousandths ? Ans. 9.7783. 

9. Add 17-^^j-, Oyx nn r, an d tttVWj fractions be¬ 
ing first changed to decimals. Ans. 23.6031. 

10. James had £ of a dollar, £ of a dollar, f of a dollar, 
iV °f a dollar, and ^ of a dollar ; change each of these 
fractions to decimals, and then find what sum he had. 

Ans. 1.525 dollar. 

SUBTRACTION OF DECIMALS. 

Lesson 79. 

1. A man who owned 4.5 acres of land, sold .875 of an 
acre ; what quantity of land had he left ? 
operation. -Explanation. We write the smaller 

4.5 number under the larger, with tenths 

.8 7 5 under tenths, &c., and subtract as in 

- vvhole numbers, for the same reason 

3.6 2 5 acres. Ans. that we add decimals as whole num¬ 

bers. The upper number is of just 
the same value as though two 0s were placed at the right 
of it. We may imagine them there, or place them there 
before subtracting if we please. 

Therefore, to subtract where there are decimals, 

Write the smaller number under the larger , with tenths 
under tenths, hundredths under hundredths , Sfc., and then 


Explain how example 1, lesson 79, is performed. 
How do we subtract when there are decimals ? 




86 


DECIMAL FRACTIONS. 


Write a denominator beneath consisting of 1 with as many 
Os annexed as there are decimals. 

The common fraction thus obtained can be changed to 
its simplest form if desired. 

7. What is the simplest form in which .75 can be ex¬ 
pressed in a common fraction ? Ans. f-. 

8. Change .00125 to a common fraction, and this frac¬ 
tion to its simplest form. Ans. 

9. What is the simplest form in which six hundred and 

twenty-five one thousandths can be expressed as a common 
fraction ? Ans. £. 

10. Express 35.36 in a whole number and a common 

fraction in its simplest form. Ans. 35^-. 

11. What common fraction in its simplest form is equal 

to .688 ? Ans. 


ADDITION OF DECIMALS. 


Lesson 78. 


1. A farmer sold at different times the following quanti¬ 
ties of hay, 14.125 tons, .75 of a ton, .5 of a ton, 1.0625 
ton, and 4 tons ; what was the whole quantity he sold ? 

Explanation. We write the quan¬ 
tities with tenths under tenths, 
hundredths under hundredths, &.C., 
and as ten one thousandths make 
one hundredth, ten one hundredths 
make one tenth, and ten tenths 
make one, it is plain that we must 
add and carry just as in whole 
numbers. 


OPERATION. 

1 4.1 25 

.7 5 
.5 

1 .0 6 2 5 
4. 

2 0.4 3 7 5 tons. Ans. 


Therefore, to add when there are decimals, 

Write the numbers with tenths under tenths , hundredths 
under hundredths , dye., and then add as in ivhole numbers; 
being careful to put a point at the left of the tenths in all cases . 

2. Add .00004, .27, 451, and 13.003. 

3. Rufus had f of a dollar, Josiah f of a dollar, Elias f 


How do we change a decimal fraction to a common fraction ? 
What can be done to the common fraction thus obtained ? 
Explain how example 1, lesson 78, is performed. 

How do we add when there are decimals ? 




DECIMAL FRACTIONS. 


87 


of a dollar, and Philip a of a dollar; change each of these 
fractions to decimals, add them, and tell what they all had. 

Ans. 2.717 dollars, nearly. 

4. Change the following fractions to decimals, and add 
them, and the whole numbers ; ’A. i, U- 

Ans. 12.1944, about. 

5. A man sold .875 of a cord of wood to one person, 

2.0625 cords to another, and .25 of a cord to a third ; how 
much did he sell to all of them ? Ans. 3.1875 cords. 

6. Add 27.01, 251, .3801, and 3.8. Ans. 282.1901. 

7. What is the sum of .75, .25, and .50 expressed in 

whole numbers and common fractions ? Ans. 1£. 

8. What is the sum of five, and seventy-five hundredths, 

four, and eight thousandths, and two hundred and three 
ten thousandths ? Ans. 9.7783. 

9. Add 17-jj^y, and fractions be¬ 
ing first changed to decimals. Ans. 23.6031. 

10. James had £ of a dollar, £ of a dollar, f of a dollar, 
iV of a dollar, and ^ of a dollar ; change each of these 
fractions to decimals, and then find what sum he had. 

Ans. 1.525dollar. 

SUBTRACTION OF DECIMALS. 

Lesson 79. 

1. A man who owned 4.5 acres of land, sold .875 of an 
acre ; what quantity of land had he left ? 
operation. J Explanation. We write the smaller 

4.5 number under the larger, with tenths 

.8 7 5 under tenths, &c., and subtract as in 

- whole numbers, for the same reason 

3.6 25 acres. Ans. that we add decimals as whole num¬ 

bers. The upper number is of just 
the same value as though two 0s were placed at the right 
of it. We may imagine them there, or place them there 
before subtracting if we please. 

Therefore, to subtract where there are decimals, 

'Write the smaller number under the larger , with tenths 
under tenths , hundredths under hundredths, Sfc., and then 


Explain how example 1, lesson 79, is performed. 
How do we subtract when there are decimals ? 




88 


DECIMAL FRACTIONS. 


subtract as in whole numbers ; being careful to put a point at 
the left of the tenths in all cases. 

2. Subtract 3.0175 from 4. 

3. Ephraim having | of a dollar, spent of a dollar ; 

change these fractions to decimals, and tell me how much 
he had left. Ans. .475 of a dollar. 

4. What is the difference between .875 of a dollar, and 

.75 of a dollar ? Ans. .125 of a dollar. 

5. A merchant who owned f of a ship, sold § of her ; 
what part of her, expressed in decimals, did he retain ? 

Ans. .1333, about. 

6. From 5 take .5. Ans. 4.5. 

7. A man who had three hundred and seventy-five one 
thousandths of a dollar, gave a boy six hundred and twenty- 
five ten thousandths of a dollar ; how much had he left ? 

Ana. .3125 of a dollar. 

8. What is the difference between 17.375 and 11.00005 ? 

Ans. 6.37495. 

« 9. Subtract .5625 from .625, and give the answer in a 

common fraction in its simplest form. Ans. jfc. 

10. Andrew promised to be gone on an errand only 
of an hour, but stayed of an hour ; what part of an hour, 
in decimals, was he gone longer than he promised to be ? 

Ans. .833 of an hour, about 


MULTIPLICATION OF DECIMALS. 

Lesson 80. 


OPERATION. 

1.3 
.0 2 


1. A man having 1.3 acre of land, sold .02 of it; how 
much was that ? 

Explanation. 2 times 1.3 are 
2.6, but as we multiply by .02, 
only TUTT °f 2 , the product is only 
Tiro - 2.6, or .026. We can 
show this result to be correct by 
another method ; thu3, changing 
1.3 and .02 to common fractions, we have and yf g-, now 
multiplying these fractions together, we get equal in 

decimals to .026. 

The product .026, must contain as many decimals as 


0 2 6 of an acre. Ans. 


Explain hbw example 1, lesson 80, is performed. 

How can we show this result to be correct by another method ? 




DECIMAL FRACTIONS. 


89 


there are decimals in 1.3 and .02 ; for .026 must contain 
as many decimals as there are Os in the denominator of 
T#®T 7 , see lesson 75, and the denominator of must 
contain as many Os as there are Os in the denominators of 
and yfy ; and the denominators of -fg and yf^ must con¬ 
tain as many Os as there are decimals in 1.3 and .02. 

So there will always be as many decimals in a product 
as in both multiplier and multiplicand. 

Therefore, to multiply numbers in which there are 
decimals, 

Multiply as in whole numbers. In the product point off 
as many decimals as there are in both multiplier and multi¬ 
plicand. 

If there be more decimals in the multiplier and multipli¬ 
cand than figures in the product, prefix Os to the product 
to supply the deficiency. 

2. What is the product of .043 by 12 ? 

3. A farmer sold at different times, 8 parcels of butter, 

each of which contained 6.25 pounds ; what was the whole 
number of pounds he disposed of? Ans. 50. 

4. What must I give for five tenths of a bushel of corn, 
if the price be seventy-five hundredths of a dollar a bushel ? 

Ans. .375 of a dollar. 

5. How much is 1.2 bushel of wheat worth, at 1.75 

dollar a bushel ? Ans. 2.1 dollars. 

6. 10 boys have .6 of n dollar apiece ; how much have 

they all ? Ans. 6 dollars. 

7. Multiply .003 by .0009. Ans. .0000027. 

8. A man had 21, .125 of a dollar ; what sum had he ? 

Ans. 2.625 dollars. 

9. A farmer sold 6§ tons of hay for 18£ dollars ; how 
much did he get for the whole in dollars and decimals ? 

Ans. 122.22 dollars, about 

10. What is the amount of .25 of .0125 ? Ans. .003125 

11. 100 men were paid .75 of a dollar apiece; what sum 

did they all get ? Ans. 75 dollars. 

How many decimal figures must the product .026 contain ? Explain 
why. 

How many decimals will there always be in a product? 

How do we multiply numbers in which there are decimals ? 

What if there be more decimals in the multiplier and multiplicand 
than figures in the product? 

8 * 



90 


DECIMAL FRACTIONS. 


DIVISION OF DECIMALS. 

Lesson 81. 

The product of the divisor and quotient is the dividend, 
so, according to lesson 80, there must be the same number 
of decimals in the divisor and quotient as in the dividend, 
and of course as many decimals in any quotient as those in 
the dividend exceed those in the divisor. 

1. 3.5 dollars are to be divided equally among 12 per¬ 
sons ; what is each one’s portion ? 

operation. Explanation. We 

1 2 ) a 3 .5 ( .2 9 of a dollar, about, cannot divide the 3 dol- 
2 4 Ans. lars by 12, so we con- 

- ' sider the sum as 35 

110 tenths of a dollar, which 

10 8 divided into 12 parts, 

- give 2 tenths for each 

2 part with 11 tenths re¬ 

mainder. We annex 0 
to the remainder 11, and considering it 110 hundredths, 
divide by 12, and get 9 hundredths, with a remainder 2, 
which being small, is neglected. The 0 annexed to 11 
must be considered as one of the decimals of the dividend, 
as it is used just as though it had been originally placed 
in it. 

2. How many times is .03 contained in 6 ? 

operation. Explanation. We first 

.0 3 ) 6.0 0(200 times. Ans. change the 6 to 600 hun- 

6 dredths, which contain 3 

—. - hundredths 200 times. 

00 

3. .01 of an acre of land is .7 of a certain quantity ; 
what is that quantity ? 


How many decimals are there in any quotient? Why ? 
Explain how example 1, lesson 81, is performed. 
Explain how example 2, lesson 81, is performed. 





DECIMAL FRACTIONS. 


91 


OPERATION. 

7 ) .0 1 0 ( .0 1 4 3 of an acre, nearly. 
7 Ans. 


3 0 
2 8 


Explanation. 7 
tenths not being 
contained in the 
tenths, we get no 
whole number ; 
not being contain¬ 
ed in the hun¬ 
dredths, we get no 
tenths, but we get 
— 1 hundredth, 4 

thousandths, and nearly 3 ten thousandths. 

Therefore, to divide one number by another, when there 
are decimals in both or either, 


20 
2 1 


Proceed as in whole numbers, annexing as many Os to the 
dividend and to each remainder, as may be necessary to 
perform the division, and to carry the operation as far as 
desired. In the quotient, poh\t off as many decimals as the 
number of decimals in the dividend, including the Os annexed 
to it, and to the remainders, exceeds the number of decimals in 
the divisor. 


If the divisor have more decimals than the dividend, 
supply the deficiency by annexing Os to the dividend before 
dividing. 

If there be not figures enough in the quotient for deci¬ 
mals, supply the deficiency by prefixing Os. 

4. Divide .05 by .003. 

5. 10 men are to receive equal shares in 27.25 dollars ; 

what is each one’s portion ? Ans. 2.725 dollars. 

6. A man bought 2.5 barrels of flour for 16 dollars ; 

what price did he give a barrel ? Ans. 6.4 dollars. 

7. If you give .75 of a dollar for .125 of a yard of cloth, 

what price do you pay a yard ? Ans. 6 dollars. 

8. .43 of a mile was divided into 6 equal parts ; what 
portion of a mile was there in each part ? 

Ans. .07167 of a mile, nearly. 


Explain how example 3, lesson 81, is performed. 

How do we divide one number by another when there are decimals 
in both or either ? 

What if the divisor have more decimals than the dividend ? 

What if there be not figures enough in the quotient for decimals? 





92 


DECIMAL FRACTIQNS. 


9. Divide .0004 by .2. Ans. .002. 

10. Divide 37 by .0403. Ans. 918.114, about. 

11. A man gave 9 cents for .75 of a pound of cheese ; 
how much must he have given for a pound ? 

Ans. 12 cents 

12. If 375 bushels of potatoes be obtained for 7.5 tons 

of hay, how many bushels should you receive for 1 ton of 
hay ? Ans. 50. 


CONTRACTIONS IN THE USE OF DECIMALS. 


Lesson 82. 

When we have either of the decimal fractions in the 
following table to multiply or divide by, it will be much 
shorter and easier to employ the corresponding common 
fraction. 


Recite the following Table. 

.0625 

is 

*• 

.125 

is 

b 

.25 

is 

h 

5 

is 

b 

.1666 &c. 

is 

b 

.333 &c. 

is 

h 

.666 &c. 

is 

f- 

.8333 &c. 

is 

*• 


1 . 


What is .333 &c. of 27.15 dollars ? 

Ans. 9.05 dollars. 

2 Multiply 45 by .1666 &c. Ans. 7.5 

3. Multiply 3.68 by .0625. Ans. .23. 

4. A man owned .5 of 432.54 acres of land; what quan¬ 
tity was that ? Ans. 216.27 acres. 

5. What is .8333 &c. of .048 ? Ans. .04 

6. Divide 7.61 by .0625. Ans. 121.76. 

7. Divide 36 by .666 &c. Ans. 54. 

8. Divide .037 by .25. Ans. .148. 

9. .125 of a hogshead of molasses cost 4 dollars ; what 

did the whole hogshead cost ? Ans. 32 dollars. 

10. Multiply 5640 by .125. Ans. 705. 

11. What sum is .25 of 11.6 dollars ? Ans. 2.9 dollars. 

12. Divide .43 by 333 &c. Ans. 1.29. 



PROMISCUOUS QUESTIONS IN FRACTIONS. 


93 


PROMISCUOUS QUESTIONS IN FRACTIONS. 
Lesson 83. 

1. The head of a fish is £ of his whole length, his body 

is f of his whole length, and his tail is 2 feet long ; what 
is the length of the fish ? Ans. 8 feet. 

2. 4 boys found 7 large apples, which they agreed to 

share equally ; what was each one’s part in a whole num¬ 
ber and a common fraction, and in a whole number and 
decimals ? Ans. If and 1.75. 

3. A man saved out of the wreck of his fortune 450 dol¬ 

lars, which was only of what he had possessed ; what 
was his fortune before his loss ? Ans. 9,000 dollars. 

4. Subtract .875 from .9. Ans. .025. 

5. Express 32f in a whole number and decimals. 

Ans. 32.667, nearly. 

6. 12 is § of what number ? Ans. 18. 

7. £ is f of what number ? Ans. §. 

8. Change 9f to an improper fraction, the denominator 

of which shall be 16 ? Ans. 

9. Now change this fraction to its simplest form. Ans. 

10. A man had f, f, and § of a dollar ; how much 

money had he ? Ans. 2 T ^ dollars. 

11. Change .15 to a common fraction, and change this 

fraction to its simplest form. Ans. jfy. 

12. 9f is £ of what number ? Ans. Ilf. 

13. 1 If is 5f times what number ? Ans. 2/ F . 

Lesson 84. 

1. If you have 9f dollars and spend 7f£ dollars, how 

much will you have left ? Ans. Iff dollar. 

2. A gentleman made a will, giving f of his estate to his 
wife, f of it to his son, and the remainder, amounting to 
2,000 dollars, to his daughter ; how much was he worth ? 

Ans. 12,000 dollars. 

3. If you give f of a dollar for f of a bushel of wheat, 
what quantity can you get for 1 dollar ? Ans. 2 bushels. 

4. A man sold § of f of a ship for 3,000 dollars ; what 
was the whole vessel worth at this rate ? 

Ans. 12,000 dollars. 

Ans. 412. 


5. What is .16667 of 2,472 ? 


94 


PROMISCUOUS QUESTIONS IN FRACTIONS. 


6. A teacher stated that § of his scholars learned to 
read and write, that f of the remainder learned geography 
and grammar, and that the rest, amounting to 5, learned 
arithmetic ; how many scholars did he have ? Ans. 60. 

7. f of a house is worth 516 dollars ; what is the value 

of the whole house ? Ans. 1,204 dollars. 

8. If you give f of a dollar for 1 bushel of corn, what 

must you give for 12^ bushels ? Ans. 9^ dollars. 

9. .375 of a quantity of goods worth 4,000 dollars was 

destroyed during a fire ; what sum will a man lose who 
owned .12 of the whole ? Ans. 180 dollars. 

10. A man sold .875 of a firkin of butter for 7 dollars ; 
what was the whole worth at this rate ? Ans. 8 dollars. 

11. If you give .625 of a dollar for 1 gallon of molasses, 
what must you give for .8 of a gallon ? Ans. .5 of a dollar. 

Lesson 85. 

To be performed in the mind. 

Ratio. 1. 15 ounces of honey were given to James and 
Henry, James receiving 3 ounces, and Henry 12 ounces ; 
what part of Henry’s share was James’ ? What part of 
James’ share was Henry’s ? 

Finding what part of 12 ounces 3 ounces are, we call 
finding the ratio of 3 to 12, and finding what part of 3 
ounces 12 ounces are, we call finding the ratio of 12 to 3. 

Therefore, to find the ratio of one number to another, 

Find what part of the second number the first is. 

2. What is the ratio of James’ share of the honey to 
Henry’s ? Of Henry’s share to James’ ? 

3. What is the ratio of 1 to 2 ? Of 2 to 4 ? Of 9 to 15 ? 
Of 100 to 3 ? Of 8 to 32 ? Of 32 to 8 ? Of 60 to 12 ? 

Proportion. 4. In example 1, what part or proportion 
of the whole 15 ounces did James get, and what proportion 
of the whole did Henry get ? 

Proportion is often used in the same sense as part, as in 
the preceding question. We also say that James and 


Finding what part of 12 ounces 3 ounces are, we call what ? Finding 
what part of 3 ounces 12 ounces are, we call what ? 

How do we find the ratio of one number to another ? 

In what sense is proportion often used ? 



PROMISCUOUS QUESTIONS IN FRACTIONS. 


95 


Henry shared the honey in the proportion of 3 to 12, and 
that Henry and James shared it in the proportion of 12 to 
3, meaning that James had T 3 2 or £ as much as Henry, 
and Henry or 4 times as much as James. 

5. A farmer mixed some rye and corn together in the 
proportion of 3 to 2 ; what part of the amount of rye was 
the amount of corn ? What part of the amount of corn was 
the amount of rye ? 

6. If the mixture had been composed of f as much corn 
as rye, in what proportion would the corn have been to 
the rye ? In what proportion would the rye have been to 
the corn ? 

7. A man in his will divided his estate between his son 
and daughter in the proportion of 7 to 5, giving the son 
3,500 dollars ; what sum did the daughter get ? 

For the Slate 

8. A man’s estate was divided among his two sons in the 
proportion of 12 to 17 ; the share of the first being 1,800 
dollars, what was the share of the second ? 

Ans. 2,550 dollars. 

Explanation. What part of the first one’s share was 
that of the second ? 

9. Four men were paid a certain sum in the proportions 
of 2, 3, 5, and 7, the first receiving 8 dollars ; what was 
given to each of the others ? 

Ans. the second had 12, the third 20, and the fourth 28 
dollars. 

Explanation. The shares of the first and second were 
in the proportion of 2 to 3, the shares of the second and. 
third were in the proportion of 3 to 5, &c. 

10. Four men, A, B, C, and D were weighed; A weigh¬ 

ed 135 pounds, and B weighed 150 ; the weights of C and 
D were in the same proportion, but that of C was 180 
pounds ; what did D weigh ? Ans. 200 pounds. 

11. A is worth 4,500 dollars, and B 6,000 dollars ; what 

part of B’s property is that of A, the fraction being changed 
to its simplest form ? Ans. A’s property is f of B’s. 


In what proportion do we say that James and Henry shared the 
honey ? In what proportion do we say that Henry and James shared it? 
What do we mean by this ? 



96 


PROMISCUOUS QUESTIONS IN FRACTIONS. 


Lesson 86. 


1. A gardener raised 80 bushels of potatoes, and a 

neighboring farmer 640 bushels ; what part of the garden¬ 
er’s quantity was the farmer’s ? Ans. 

2. How many times the gardener’s quantity was the 

farmer’s ? Ans. 8 times. 

3. 80 dollars were contributed to relieve a poor woman; 

what proportion of the whole did a man give who put in 5 
dollars ? Ans. y 1 ^. 

4. What part of £ is % ? Ans. §. 

Explanation. To obtain the answer by the rule, we 


make £ the denominator, thus, 


i. 

v 


now 


dividing the nu¬ 


merator by the denominator, we get f. 

5. 6£ dollars are given to John, Samuel, and William ; 
John received 2£ dollars, Samuel 1|£ dollars, and William 
2^£ dollars; what proportion of the whole did each receive ? 

Ans. John received Samuel and William T ^-. 

6. A owns .0625 of a ship, and B .25 of her ; what part 

of B’s share is A’s ? Ans. £, or .25. 


Average. 7. A man raised 16 bushels of wheat on one 
acre of land, and 19 bushels on another acre ; what was 
the average number of bushels he raised to the acre on the 
two pieces ? Ans. 17£ bushels. 

Explanation. Average signifies equal division ; to aver¬ 
age means to divide equally. 

8. The wages of 5 laborers are as follows ; the first 
receives f of a dollar a day, the second £ of a dollar, the 
third 1 dollar, the fourth 1^ dollar, and the fifth dollar: 
what wages do they get on an average ? Ans. 1^ dollar. 

9. A man measured a certain distance 3 times, the first 

time he made the distance 4,712.25 feet, the second time 
4,710.85 feet, and third time, 4,713.11 feet; how far did 
he make it on an average ? Ans. 4,712.07 feet. 

10. If you sell 11 bushels of corn for 9f dollars, 7 
bushels for 6 T 9 F dollars, and 7 bushels for 7-^f dollars, what 
is the average price you get a bushel. Ans. §A of a dollar. 

11. What is the average of the following numbers, 1, 2, 

5, 17, and 34 ? Ans. 11.8. 


What does average signify r What does to average mean i 



FEDERAL MONEY. 


97 


FEDERAL MONEY. 

Lesson 87. 

In the United States money is reckoned in dollars, 
dimes, cents, and mills. The dollars are considered as 
whole numbers, and the dimes, cents, and mills as deci¬ 
mals ; the dimes being tenths, the cents hundredths, and 
the mills thousandths of a dollar. 

10 mills make 1 cent. 

10 cents make 1 dime. 

10 dimes or 100 cents make 1 dollar. 

The dollar mark $, placed before any figures, shows 
that they express this money, which is called Federal 
Money ; thus, 

8 5.257 signifies 5 dollars, 2 dimes, 5 cents, and 7 mills. 
It is not customary, however, to use the word dime, but 
dimes are expressed in cents ; thus, 

8 5.257 is read, 5 dollars, 25 cents, and 7 mills 

$5.20 is read, 5 dollars and 20 cents. 

8 35.05 is read, 35 dollars and 5 cents. 

8 435.207 is read, 435 dollars, 20 cents, and 7 mills. 


Read the following sums from your slate. 


1. 

... 8 4.08 1 

1 6 * 

...8 

500.55 1 

1 n - 

...8 

233.00 

2. 

...8 .234 

7. 

...8 

85.307 

12. 

...8 

.005 

3. 

...8 .40 

8. 

...86 

,334.093 

13. 

...8 

.265 

4. 

...828.004 

9. 

... 8 

999.999 

! 14. 

...813,760.08 

5. 

...813.50 1 

1 10. 

...8 

7.001 1 

1 15. 

...8 

.06 


Write the following sums on your slate in figures. 

1. Thirty-five cents. 

2. Seventeen dollars three cents and five mills. 

In what is money reckoned in the United States? What are consid¬ 
ered as whole numbers in this money? What are considered as deci¬ 
mals ? What are the dimes ? Cents ? Mills ? 

How many mills make one cent ? Cents one dime ? Dimes or cents 
one dollar ? 

What shows that figures express this money ? What is this money 
called ? 

What does $ 5.257 signify ? Is it customary to use the word dime ? 
How are dimes expressed ? 

How do you read $ 5.257 ? $ 5.20 ? $ 35.05 ? $ 435.207 ? 

9 







98 


FEDERAL MONEY. 


3. Two hundred thousand dollars and twelve cents. 

4. Seven hundred and fifteen dollars sixteen cents and 
five mills. 

5. Eighteen dollars and seven mills. 

6. Twenty-five dollars and fifty cents. 

7. Eighty cents. 

8. Five hundred and seven dollars thirty cents and one 
mill. 

9. Thirteen cents and two mills. 

10. Six cents. 

11. Eighty dollars. 

12. Two mills. 

13. Ninety dollars thirty-seven cents and one mill. 

14. Five cents and three mills. 

15. Sixteen hundred and twelve dollars. 

16. Five thousand one hundred and fifty dollars and 
fifty cents. 

Lesson 88. 

To be performed in the mind. 

1. How many cents are there in $ 1 ? In $ 5 ? In $ 25 ? 
In $28 ? In $43 ? 

2. A trader had 600 cents ; how many dollars were they 

worth ? How many dollars are 400 cents worth ? 1,200 

cents ? 300 cents ? 250 cents ? 

3. How many mills are there in $ .035 ? In $ .20 ? In 
$13 ? In $7 ? In $24 ? 

For the Slate. 

4. A man exchanged $42 for dimes, or ten cent pieces ; 
how many did he receive ? How many cents could he have 
got for the $ 42 ? How many mills were there in the $ 42 ? 
How many dimes are there in $ 42.20 ? How many cents ? 
Mills ? How many dimes are there in $ 42.25 ? How 
many cents ? Mills ? How many dimes are there in 
$ 42.259 ? How many cents ? Mills ? 

5. How many cents are there in 1,600 mills ? How 
many dimes ? Dollars ? How many cents are there in 
1,650 mills ?. How many dimes ? Dollars ? 

6. A man received 2,317 cents for a debt ; how many 
dollars did he get ? 

7. How many dimes, or ten cent pieces, are 15 ten dol¬ 
lar bills worth ? How many cents are they worth ? 


FEDERAL MONEY. 


99 


8. Change $ 5,827.37 to cents. 

9. Change 83,254 mills to dollars. 

10. A grocer wishes to get $250, in cents, for change ; 
how many cents can he obtain for that sum ? 

Lesson 89. 

For the Slate. 

As Federal Money is composed of dollars and decimals 
of a dollar, we must add, subtract, multiply, and divide in 
it, as in Decimal Fractions. 

1. A farmer owes the following sums to different persons; 

$ 1,325.043, $2,875, $835, $ 17.50, and $.375 ; what is 
the whole amount of his debts ? Ans. $2,180,793. 

2. If a man gives you his note for $ 180, and afterwards 
pays you $ 2.50, how much will he then owe you ? 

Ans. $ 177.50. 

3. A tailor bought 27f yards of broadcloth at $ 4.50 a 

yard ; how much must he pay for it ? Ans. $ 125.10. 

Explanation. First change to decimals. 

4. A man paid $45.25 for 35 bushels of wheat ; what 

price did he pay a bushel ? Ans. $ 1.293, nearly. 

5. A farmer sold 4 cows at $ 13.75 apiece, 3 calves at 
$ 4.33^- apiece, 33 bushels of potatoes at $ .25 a bushel, 
and 2il§ pounds of cheese at $ .125 a pound ; how much 
money did he sell the whole for ? Ans. $ 102.708, about. 

Explanation. Observe that $ .25 is £ of a dollar, and 
$ .125, |of a dollar. 

6. A laborer earns $1.25 a day, and spends $.41f a 

day ; how much will he save of what he obtains for 42 
days’ labor ? Ans. $35. 

7. 6 men owned equal- shares in 13 barrels of pork, 

which they sold at $ 26 a barrel ; what sum out of the 
proceeds must each receive ? Ans. $ 56.333, about. 

8. If salt be worth $.75 a bushel, and corn $ 1.12£ a 
bushel, how much corn must I give for 83 bushels of salt ? 

Ans. 554 bushels. 

9. Add $3,777.04, $ 12.057, and $.T2£. Ans. 3,789.222. 

IQ. A trader owes $ 4,327.17, he has a house which will 

bring $ 2,500, $ 1,304.07 in cash, 4 hogsheads of molasses. 


How do we add. subtract., multiply, and divide in Federal Money ? 
Why ? 




100 


FEDERAL MONEY. 


each of which contains 100 gallons, worth $.31£ a gallon, 
sundry other goods worth $245, and various persons owe 
him $1,250 ; how much property will he have after paying 
his debts ? Ans. $1,096.90. 


Lesson 90. 

The coins used in the United States, are the eagle , or 
ten-dollar piece, the half eagle , and the quarter eagle , which 
are of gold ; the dollar, the half dollar, the quarter of a 
dollar, the eighth of a dollar, the sixteenth of a dollar, the 
dime or ten-cent piece, and the half dime or five-cent piece, 
which are of silver ; and the cent, which is of copper. All 
of these pieces are American coins, except the dollar and 
quarter of a dollar, which are most commonly Spanish, and 
the eighth and sixteenth of a dollar, which are always 
Spanish coins. 

The eighth of a dollar, or 12£ cent piece, is called a 
ninepence in New England, a shilling in New York and 
some other States, and a levy or an elevenpenny bit, or sim¬ 
ply a bit, in Pennsylvania and in some other States. 

The sixteenth of a dollar, or 6£ cent piece, is called a 
fourpence halfpenny in New England, a sixpence in New 
York and in some other States, a fip or a Jivepenny bit in 
Pennsylvania, and in some other States, and a pecune or 
a picayune in some of the southwestern States. 

The custom of dividing things into eighths and six¬ 
teenths, and the prevalence of eighths and sixteenths of a 
dollar, render it convenient to be familiar with the value of 
different numbers of these pieces. 


Recite the following Table. 

1 eighth of a dollar is. 12£ cents. 

2 eighths of a dollar are £ of a dollar or .. 25 cents. 

3 eighths of a dollar are. 37£ cents. 


What coins are used in the United States, and of what metals are 
they composed ? Which are American coins, and which Spanish ? An 
eagle being worth $10, what is a half eagle worth ? A quarter eagle ? 
How many cents is a half dollar worth? A quarter of a dollar? An 
eighth of a dollar ? A sixteenth of a dollar ? 

By what names is the eighth of a dollar, or 12£ cent piece called in 
different places ? By what names is the sixteenth of a dollar, or 6£ cent 
piece called in different places ? 

What is said of the custom of dividing things into eighths and six- 
teenths ? 





FEDERAL MONEY. 


10! 


4 eighths of a dollar are £ of a dollar or .. 50 cents. 

5 eighths of a dollar are. 62£ cents. 

6 eighths of a dollar are f of a dollar or .. 75 cents. 

7 eighths of a dollar are . ... 87^- cents. 

8 eighths of a dollar are 1 dollar or . 100 cents. 


To be performed in the mind. 

1. How many eighths of a dollar are 2 sixteenths ? 
What then is the value, in cents, of 2 sixteenths of a dollar ? 

2. What is the value, in cents, of 3 sixteenths of a dollar ? 

3. How many eighths of a dollar are 4 sixteenths ? 
What part of a dollar are 4 sixteenths ? 

4. What is the value, in cents, of 5 sixteenths of a dollar ? 

5. How many eighths of a dollar, and how many cents 
are 6 sixteenths worth ? 

6. How many cents are 7 sixteenths worth ? 

7. How many eighths of a dollar are 8 sixteenths worth ? 
What part of a dollar do 8 sixteenths make ? 

8. How many cents are 9 sixteenths worth ? 

9. How many eighths, and how many cents are 10 six¬ 
teenths worth ? 

10. How many cents are 11 sixteenths worth ? 

11. How many eighths are 12 sixteenths worth ? What 
part of a dollar do 12 sixteenths make ? 

12. How many cents are 13 sixteenths worth ? 

13. How many eighths, and how many cents are 14 six¬ 
teenths worth ? 

14. How many cents are 15 sixteenths worth ? 

15. How many eighths are 16 sixteenths worth ? What 
part of a dollar do 16 sixteenths make ? 

Lesson 91. 

To be performed in the mind. 

1. James bought 5 pears at 3 cents apiece, and paid 1 
eighth and 1 sixteenth for them ; how much change must 
he receive back ? If he had paid 2 eighths, how much 
change should he have received back ? 

2. If you buy a penknife for $.62£, and have 1 half dol¬ 
lar and 4 sixteenths in your purse, how will you contrive 
to pay or make change ? 

3. A man gave 1 dollar in payment for a book, and 
received 3 eighths back ; what was the price of the book, 
in cents ? 


9 * 





102 


FEDERAL MONEY. 


4. If you buy a quire of paper for $.25, some quills for 
1 sixteenth, and an inkstand for 3 eighths, what sum must 
you pay for the whole ? How many sixteenths ? 

5. At an auction, A bid six eighths for a book, and B 13 
sixteenths ; what number of cents did each bid ? How much 
more did B bid than A ? 

6. If you should buy a handkerchief in New York for 4 
shillings, how many New England ninepences would you 
give in payment ? What sum would you have given if the 
price had been 4 shillings and sixpence ? 

7. If a hackman demand 6 shillings for carrying you 
from the foot of Barclay street, in New York city, to the 
Astor House, how many cents should you pay him ? 

8. If you buy articles of a trader in Philadelphia to the 
amount of 3 levies and 1 tip, how many cents should you 
pay him ? 

9. A man was charged 5 picayunes for some luncheon in 
New Orleans, and paid with a 3-dollar bill ; how many 
dollars and cents should he receive back ? 

10. If you carry a pocket full of ninepences and four- 
pence-ha’pennies from Boston to New York, what will you 
call them there ? What will you call them in Philadelphia ? 

Lesson 92. 

In New England 16§ cents are called a shilling, 8^ 
cents, or ^ of a shilling, are called sixpence, and 4£ cents, 
or £ of a sixpence, are called threepence. There are no 
coins of these values, but prices are often named in these 
shillings, sixpences, and threepences. 

Recite the following Table. 


1 shilling is. 16§ cents. 

1 shilling and sixpence are.. 25 cents. 

2 shillings are. 33£ cents. 

2 shillings and sixpence are. 4l-§ cents. 

3 shillings are... 50 cents. 

3 shillings and sixpence are. 58£ cents. 

4 shillings are. 66f cents. 

4 shillings and sixpence are . 75 cents. 


What in New England are called a shilling ? Sixpence ? Three¬ 
pence ? Are there any coins of these values ? What is often named in 
these shillings, &c. ? 







FEDERAL MONEY. 


103 


5 shillings are. 

5 shillings and sixpence are. 

6 shillings are ... 


83£ cents. 

91 § cents. 

100 cents or 1 dollar 


To be performed in the mind. 


1. If you buy 3 yards of calico in Boston, at 1 shilling a 
yard, what part of a dollar must you give for your purchase? 

2. A man bought a knife in Providence for 4 shillings 
and sixpence ; how many cents did he give for it ? How 
many ninepences ? How many dimes ? 

3. A man pays 15 shillings a week for his board in 
Portsmouth ; how many dollars and cents is that price ? 
Explanation. As 6 shillings make 1 dollar, find how many 
times 6 shillings there are in 15 shillings, and how many 
shillings over. 

4. A laborer in Boston has 7 shillings and sixpence a 
day ; how many dollars and cents will he get in 4 days ? 

5. If a laborer gets 6 shillings and ninepence for a da^ ’s 
work, and pays 2 shillings and sixpence out of it for a 
handkerchief, how many cents will he have left ? 

6. A young man bought a pair of gloves in Lowell for 9 
shillings ; how many dollars and cents did he pay ? If he 
had paid 8 shillings for them, how many dollars and cents 
would that sum have been ? 

7. If you buy a pair of boots in Portland for 16 shillings, 
how many dollars and cents must you pay for them ? 

8. If you buy a pair of stockings for 2 shillings and six¬ 
pence, and give 4 ninepences in payment, how much 
change must you receive back ? 

9. A farmer bought 3 gallons of molasses at 2 shillings 
and threepence a gallon; what must he pay for it in dollars 
and cents ? 

10. What number of cents are 5 shillings and three¬ 

pence ? 7 shillings and threepence ? 9 shillings and three¬ 
pence ? 12 shillings and threepence ? 




104 


COMPOUND NUMBERS. 


COMPOUND NUMBERS. 

Lesson 93. 

■ 

Quantities are divided into parts of different sizes, for 
the purposes of traffic and convenience ; thus we measure 
salt in bushels, pecks, and quarts, 8 quarts making 1 peck, 
and 4 pecks 1 bushel. The parts or divisions are called 
denominations. Numbers expressing a quantity in different 
denominations are called compound numbers. Those here¬ 
tofore used may be called simple numbers. 

Recite the following Tables. 

AVOIRDUPOIS WEIGHT. 

Avoirdupois Weight is the, common weight, and is used 
in weighing all common and coarse articles. 

16 drams, sign dr..... .make 1 ounce,.sign, oz. 

16 ounces.make 1 pound,.sign, lb. 

28 pounds.make 1 quarter,.sign, qr. 

4 quarters, or 112 lbs. make 1 hundred weight, sign, cwt. 

20 hundred weight . .. .make 1 ton,.sign, T. 

This is the old manner of weighing. At present it is 
usual to buy and sell by the 100 pounds, and when the 
term ton is employed, it generally means 2,000 pounds. 

TROY WEIGHT. 

Troy Weight is used in weighing gold and silver. 

24 grains, sign gr.make 1 pennyweight, ..sign, pwt. 

20 pennyweights.make 1 ounce,.sign, oz. 

12 ounces.make 1 pound,.sign, lb. 


How are quantities divided ? For what purposes ? Give an example. 
What are called denominations ? Compound numbers ? Simple num¬ 
bers P 

What is Avoirdupois Weight, and for what is it used ? 

What is it usual to buy and sell by, at present, in Avoirdupois 
Weight? What does the term ton generally mean ? 

For what is Troy Weight used ? 















COMPOUND NUMBERS. 


105 


apothecaries’ weight. 

Apothecaries’ Weight is used in compounding medicines, 
but not in selling them. They are sold by Avoirdupois 
Weight. 

20 grains, sign gr.make 1 scruple,... .sign. B. 

3 scruples.make 1 dram,.sign, b. 

8 drains.make 1 ounce,.sign, §. 

12 ounces.make 1 pound,.sign, [fr. 

The pound and ounce in Apothecaries’ Weight are the 
same as the pound and ounce in Troy Weight. 175 pounds 
Troy Weight, are equal to 144 pounds Avoirdupois Weight. 
There are 7,000 grains in 1 pound Avoirdupois, and 5,760 
grains in 1 pound Troy. 


Lesson 94. 

LONG MEASURE. 

Long Measure is used in measuring distances. 

12 inches, sign in.. make 1 foot,.sign, ft. 

3 feet.make 1 yard,.sign, yd. 

16£ feet.make 1 rod, perch, or pole, sign, rod. 

40 rods.make l furlong,.sign, fur. 

8 furlongs.make 1 mile,.sign, m. 

The following measures are used at sea. 

6 feet.make 1 fathom,.sign, fath. 

3 miles.make 1 league,.sign, lea. 


For what is Apothecaries’ Weight used P 

What is the difference between the pound and ounce Apothecaries’ 
Weight, and the pound and ounce Troy Weight ? How many pounds, 
Troy Weight, are equal to 144 pounds, Avoirdupois Weight ? How many 
grains are there in 1 pound Avoirdupois Weight ? In 1 pound Troy 






















JOB 


COMPOUND NUMBERS. 


LAND, OR SQUARE MEASURE. 

Land, or Square Measure, is used in measuring land or 
any surface. 

Figure 1. 

1 yard or 3 feet. Explanation of this measure. A yard 
is 3 feet long, but a square yard is a 
surface 3 feet long and 3 feet wide, as 
figure 1. By examining this figure, 
we see that a space 1 foot wide and 3 
feet long, contains 3 square feet ; that 
a space 2 feet wide and 3 feet long, 
contains 2 times 3, or 6 square feet ; 
and that a space 3 feet wide and 3 feet long, contains 3 
times 3, or 9 square feet. 

It is also evident from this, that a square foot, being 12 
inches long and 12 inches wide, contains 12 times 12, or 
144 square inches, &.c. 

144 square inches make 1 square foot,.sign, sq. ft. 

9 square feet.. .make 1 square yard, ... .sign, sq. yd. 

272£ square feet..make 1 square rod,.sign, sq. rod. 

40 square rods . .make 1 quarter of an acre, sign, qr. 

4 quarters . .. . .make 1 acre,...sign, A. 

640 acres.make 1 square mile, ... .sign, sq. mile 

CUBIC, OR SOLID MEASURE. 

Cubic, or Solid Measure, is used in measuring bodies, 
or in finding the capacity of rooms, boxes, &c. 

Explanation of this measure. A cubic yard is a body 
3 feet long, 3 feet wide, and 3 feet high, as figure 2. 


For what is Land, or Square Measure used ? 

What is a square yard ? Explain why a square yard contains 9 square 
feet. 

How many square inches then does a square foot contain ? 

For what is Cubic, or Solid Measure used ? 

What is a cubic yard ? 














COMPOUND NUMBERS. 


107 


Figure 2. 



144, or 1728 ctibic inches. 


By examining this figure, we see 
that the surface of the top con¬ 
tains 9 square feet ; so if we 
take a piece off from the top 1 
foot thick, we shall get 9 cubic 
feet; if we take a piece off from 
the top 2 feet thick, we shall 
get 2 times 9, or 18 cubic feet; 
and if we take up the whole, 
which is 3 feet thick, we shall 
get 3 times 9, or 27 cubic feet. 

a cubic foot contains 12 times 


1728 cubic inches..... .make 1 cubic foot, sign, cubic ft. 

27 cubic feet.make 1 cubic yard, sign, cubic yd. 

16 cubic feet.make 1 foot of wood, sign, ft. 

8 feet of wood.make 1 cord of wood, sign, C. 

50 cubic feet of timber ,) , , , m 

whether hewed or round > ma e on ’. S1 g n > 

40 cubic feet of round timber, when £ is allowed for waste 
from knots, crooks, &c. make 1 ton, sign, T. 


Lesson 95. 

DRY MEASURE. 

Dry Measure is used in measuring'grain, fruit, potatoes, 
salt, coal, and other dry articles. 

2 pints, sign pt.make 1 quart,.•sign, qt. 

8 quarts.make 1 peck,.sign, pk. 

4 pecks.make 1 bushel,.sign, bu. 

8 bushels.make 1 quarter,.sign, qr. 

36 bushels.make 1 chaldron,.sign, chal. 

A bushel contains 2,150.4 cubic inches. 


Explain why a cubic yard contains 27 cubic feet. 

How many cubic inches then does a cubic foot contain ? 
When do 40 cubic feet make 1 ton ? 

For what is Dry Measure used ? 

How many cubic inches does a bushel contain ? 
































108 


COMPOUND NUMBERS. 


BEER MEASURE. 

Beer Measure is used in measuring ale, beer, and milk. 

2 pints, sign pt.make 1 quart,.sign, qt. 

4 quarts.make 1 gallon,.sign, gal. 

9 gallons..make 1 firkin,.sign, fir. 

2 firkins.make 1 kilderkin,.sign, kil. 

2 kilderkins, or 36 gals, make 1 barrel,.sign, bl. 

barrel, or 54 gals., .make 1 hogshead.sign, hhd. 

2 hogsheads .make 1 butt,.sign, bt. 

WINE MEASURE. 

Wine Measure is used in measuring all liquors except 
ale, beer, and milk. 

4 gills, sign gi.make 1 pint,.sign, pt. 

2 pints.make 1 quart,.sign, qt. 

4 quarts.make 1 gallon,.sign, gal. 

31^ gallons.make 1 barrel,.sign, bl. 

2 barrels, or 63 gals, .make 1 hogshead,.sign, hhd. 

2 hogsheads.make 1 pipe,.sign, p. 

2 pipes.make 1 tun,.sign, T. 

42 gallons.make 1 tierce,...sign, tier. 

2 tierces, or 84 gals, .make 1 puncheon,.... .sign, pun. 

In the United States £ of a peck, sometimes called the 
Dry gallon, contains 268.8 cubic inches ; the Beer gallon 
contains 282 cubic inches, and the Wine gallon 231 cubic 
inches. In Great Britain, since 1826, the Imperial gallon , 
containing 277.274 cubic inches, has been used in place of 
the Dry, Beer, and Wine gallons. 

The casks, called hogsheads, are of various capacities, 
but usually contain more than 100 gallons. 


For what is Beer Measure used P 
For what is Wine Measure used ? 

How many cubic inches does each of the different kinds of gallons 
used in the United States, contain ? What is now used in Great Britain 
in place of the Dry, Beer, and Wine gallons? 

How much do the casks, called hogsheads, usually contain? 































COMPOUND NUMBERS. 


109 


Lesson 96. 

OF TIME. 

60 seconds, sign.sec.make 1 minute, ....sign, min. 

60 minutes.make 1 hour, ......sign, h. 

24 hours.make 1 day, .......sign, d. 

7 days. make 1 week,...... sign, w. 

365 days...make 1 year,.sign, yr. 

There are 12 calendar months in a year, each of which 
contains 31 days, except April, June, September, and No¬ 
vember, which have 30 days, and February, which has 28. 
There are really 365 days, 5 hours, 48 minutes, and 49.7 
seconds in a year, or nearly 365£ days ; so one year in 
four, we give to February 29 days, thereby making the 
year consist of 366 days ; such a year is called leap year. 
As this allowance is a little too much, we omit 3 leap years 
in 400 years. Any year, at the end of a century, that can 
be divided by 400 without a remainder, is leap year, as 
1200, 1600, 2000. Any other year that can be divided by 4 
without a remainder, is also leap year, as 1836, 1840, 1844. 

4 weeks are sometimes called a month. 

DIVISION OF CIRCLES. 

The circumference of every circle, whether great or 
small, is considered to be divided into 360 equal parts, call¬ 
ed degrees. 

60 seconds, sign " make 1 minute,.sign, '. 

60 minutes.make 1 degree,.sign, °. 

360 degrees ...-.make 1 circumference, sign, circum. 

MISCELLANEOUS TABLE. 

12 things.make I dozen,.sign, doz. 

12 dozen.make 1 gross. 

12 gross .. .make 1 great gross. 

How many calendar months are there in a year ? How many days 
does each month contain ? What is the real length of the year? What 
then is done one year in four? What is such a year called? How 
many leap years do we omit in 400 years ? Why ? What years are leap 
years ? What are 4 weeks sometimes called ? 

How is the circumference of every circle, whether great or small, 
considered to be divided ? 

10 




















110 REDUCTION OF COMPOUND NUMBERS. 

20 things ..make 1 score. 

24 sheets of paper make 1 quire. 

20 quires.make 1 ream. 

6 points.make 1 line, ( used in measuring 

12 lines.make 1 inch, ( clock pendulums. 

. . . , , , j ( used in measuring 

4 lnches . make 1 hand ’ \ the height of horses. 

112 pounds....make 1 quintal of fish. 

200 pounds.make 1 barrel of beef or pork. 

19G pounds.make 1 barrel of flour. 


REDUCTION, OR CHANGE OF FORM OF COMPOUND NUMBERS. 

Lesson 97. 

To be performed in the mind. 

1. How many ounces of butter are there in 2 pounds 3 
ounces ? 

2. How many yards, and how many odd feet are there in 
a pole 10 feet long ? 13 feet long ? 

3. How many quarts are there in 2 bushels ? In 1 bushel 
2 pecks ? 

4. A grocer on the 4th of July, retailed 10 quarts 1 pint 
of wine ; how many gallons and odd quarts and pints did 
he sell ? 

5. If you are 3 hours and 5 minutes getting your lesson, 
how many minutes are you ? 

6. How many weeks are there in 28 days ? In 32 days ? 
In 37 days ? 

7. At 1 dollar an ounce, how much are 2 pounds of old 
silver worth ? 3 pounds ? 5 pounds ? 1 pound 3 ounces ? 

8. How many gallons of molasses are there in 16 pints ? 
In 76 pints ? In 15 quarts ? In 35 quarts ; 

9. How many square inches are there in 2 square feet 
12 square inches ? 

10. A man bought 48 pecks of oats at different times, 
and gave $6 in payment ; what price a bushel did he give ? 

For the Slate. 

11. A farmer had 5 cwt. 3 qrs. 12 lbs. of cheese ; how 
many pounds had he ? 










REDUCTION OF COMPOUND NUMBERS. 


Ill 


OPERATION, 
cwt. qrs. 

5 3 

4 

20 

Add 3 qrs. 


Add 



lbs. 

12 


6 5 6 lbs. Ans. 


Explanation. There are 4 qrs. 
in 1 cwt., so to get the number of 
qrs. we multiply the 5 cwt. by 4, 
and add the 3 qrs. to the product. 
There are 28 lbs. in a qr., so to 
get the number of lbs. we multi¬ 
ply the 23 qrs. by 28, and add the 
12 lbs. to the product. 


Therefore, to change or reduce a quantity to a lower 
denomination, 


Multiply the highest denomination in it by so many of the 
next lower as make one of this highest } and add to the product 
the number in the lower denomination ; multiply the result in 
the same way t and so on until the quantity be brought into a 
denomination as low as desired. 


12. A farmer having 656 lbs. of cheese, wished to know 
how many cwt. he had ; how many had he ? 


OPERATION. 

4 


28)656(23(5 cwts. 
56 20 


9 6 3 qrs. 

8 4 

1 2 lbs. 

5 cwts. 3 qrs. 12 lbs. Ans. 


Explanation. There being 
28 lbs. in 1 qr., we divide 656 
lbs. by 28, and get 23 qrs. and 
12 lbs. over. There being 4 
qrs. in 1 cwt., we divide 23 
qrs. by 4, and get 5 cwt. and 
3 qrs. over. So 656 lbs. are 
5 cwt. 3qrs. 12 lbs. 


Therefore, to change or reduce a quantity to a higher 
denomination, 


Explain how example 11, lesson 97, is performed. 

How do we change or reduce a quantity to a lower denomination? 
Explain how example 12, lesson 97, is performed. 





112 REDUCTION OF COMPOUND NUMBERS. 

Divide by as many as it takes to make one of the next 
higher denomination; divide the quotient in the same way, 
and so on until the quantity be brought into a denomination 
as high as desired. 

The operations under either of the two preceding rules 
are proved by reversing them. Thus the operation in ex¬ 
ample 11 is reversed in example 12, and the operation in 
example 12 is reversed in example 11 
Note. Each example should now be proved. 

Lesson 98. 

For the Slate. 

1. How many tons are there in 36,000 ounces ? 

Ans. 1 T. 10 lbs 

2. How many pounds are there in 5 tons ? Ans. 11,200. 

3. A jeweller has 4,312 pennyweights of gold, in vari¬ 
ous pieces ; how many pounds has he ? 

Ans. 17 lbs. 11 oz. 12 pwts. 

4. How many grains of silver are there in 6 ounces ? 

Ans. 2,880. 

5. How many grains of ipecacuanha are there in 21b 4§ 

25 09 1 gr. ? Ans. 13,561. 

6. How many ounces of calomel are there in 640 grains ? 

Ans. IS 23 29. 

7. A surveyor has a chain containing 100 links, each 

link being 7.92 inches long ; how many rods are there in 
the chain ? Ans. 4. 

8. How many fathoms are there in 2 leagues ? 

Ans. 5,280. 

9. A man sold a house lot 6 rods long and 4 rods wide,' 
at $.0625 a square foot ; what sum did it bring ? 

Ans. $408,375. See lesson 82, Decimal Fractions. 

10. How many square yards of carpeting will cover a 

floor 18 feet long, and 16 feet wide ? Ans. 32. 

Lesson 99. 

1. How many cubic inches are there in a block of wood 
3 feet long, 2 feet wide, and 1 foot thick ? Ans. 10,368. 


How do we change or reduce a quantity to a higher denomination ? 
How are the operations under either of the two preceding rules proved ? 
Give some examples. 



REDUCTION OF COMPOUND NUMBERS j j 3 

2. How many cords are there in a pile of wood 37 feet 
long, 4 feet wide, and 5 feet high ? 

Ans. 5 C. 6 ft. 4 cubic ft. 

3. A trader bought some beans at $.50 a peck ; ,what 
sum must he pay for 12 bushels and 3 pecks ? Ans. $25.50 

4. How many chaldrons are there in 112 bushels of 

Richmond coal ? Ans. 3 chal. 4 bu. 

5. How many pints of milk are there in 12 gallons 3 

quarts 1 pint, and how much is the whole worth, at 2 cents 
a pint ? Ans. 103 pts., and it is worth $2.06. 

6. How many firkins of beer are there in 1,313 quarts ? 

Ans. 36 fir. 4 gals. 1 qt. 

7. If a man drinks 2 quarts of wine a day, how long will 
2 tierces 5 gallons and 2 quarts last him ? Ans. 179 days. 

8. How many hogsheads of molasses are there in 2,217 

quarts ? Ans. 8 hhds. 50 gals. 1 qt. 

9 How many weeks are there in 1,000,000 seconds ? 

Ans. 1 w. 4 d. 13 h. 46 m. 40 sec. 

10. 20 cubic feet of water run over a mill-dam in a 
second ; how many cubic feet will pass over the dam in 1 
week 4 days 7 hours 0 minutes 48 seconds ? 

Ans. 19,512,960. 


Lesson 100 . 

1. In 17° 26' 14" how many seconds ? Ans. 62,774. 

2. How many degrees are there in 5,700" ? Ans. 1° 35'. 

3. How many days were there between the time of the 
Declaration of Independence, July 4th, 1776, and the time 
of the settlement of a general peace, January 20th, 1783 ? 

Ans. 2,391. 

Explanation. 1780 can be divided by 4 without a re¬ 
mainder. 

4. 4 weeks are often called a month ; how many of such 

months are there in 67 days ? Ans. 2 mo. 1 w. 4 d. 

5. A man dug a cellar 36 feet long, 24 feet wide, and 6 

feet deep ; how many cubic yards of earth did he take out 
of it ? Ans. 192. 

6. A farmer sold some oak wood at $ 1 a foot ; how 

much did he get for 3 cords and 7 feet ? Ans. $31. 

7. How many rods are there in 1 mile and 33 rods ? 

Ans. 353. 

8. How many leagues and fathoms are there in 17 miles 

and 45 fathoms ? Ans. 5 lea. 1,805 fath. 

10 * 


114 


REDUCTION OF COMPOUND NUMBERS. 


9. How many square yards of cloth are there in a piece 
28 yards long, and 6 quarters, that is, f- wide ? Ans. 42. 

10. A gold beater has 453,778 square inches of gold 
leaf; how many square yards are there in that quantity ? 

Ans. 350 sq. yds. 1 sq. ft. 34 sq. in. 

Lesson 101 . 

1. How many tons of flour are there in 3,600 barrels 
and 425 half barrels ? Ans. 333 T. 11 cwt. 3 qrs. 14 lbs, 

2. What sum are 3 tons 2 quarters and 17 pounds of 

butter worth at $16§ a pound ? Ans. $ 1,132.16§. 

3. A silversmith has 7 pounds of silver in an ingot; how 
many pennyweights are there in the ingot ? Ans. 1,680. 

4. A retailer sells beans at $.12^- a quart; what sum will 

he get for 5 bushels ? Ans. $ 20. 

5. How many bushels will a box 5 feet square hold ? 

Ans. 100.446 bushels, about. 

Explanation. Find the number of cubic inches in the 
box, and divide by the number of cubic inches in a bushel, 

6. A man retailed 35,217 pints of beer ; how many 
hogsheads was that quantity ? Ans. 81 hhds. 28 gals. 1 pt. 

7. How much are 4 hogsheads of beer worth at $.50 a 

gallon ? Ans. $ 108. 

8. How many gallons of molasses are there in 16 casks, 

called hogsheads, each of which holds 108 gallons, and in 
5 barrels ? Ans. 1,885£. 

9. How much are 5 puncheons of rum worth at $ 1 a 

gallon ? Ans. $420. 

10. A number of silver spoons weigh 428 pennyweights; 
how many pounds of silver do they contain ? 

Ans. 1 lb. 9 oz. 8 pwts. 

Lesson 102 . 

To be performed in the mind. 

1. What part of a bushel are 3 pecks ? 2 pecks ? 1 peck ? 

2. A man bought .5 of a pound of butter ; how many 
ounces did he get ? How many ounces 'would he have got 
had he bought .25 of a pound ? 1.5 pound ? 

3. Two boys had a stick of candy 1 foot in length ; they 
broke it so that the largest boy got a piece 8 inches long ; 
what part of the whole was that ? 


REDUCTION OF COMPOUND NUMBERS. \\Q 

4. What part of a week are 3 days ? 5 days ? 11 days ? 

5. How many hours are there in 4 of a day ? Jn 4 of a 
day ? In $ of a day ? 

6. What part of a gallon are 1 quart and 1 pint ? 

7. If I buy .6 of a ton of hewed timber, how many cubic 
feet do I get ? 

8. How many feet of wood are there in £ of a cord ? In 
& of a cord ? In .5 of a cord ? In .25 of a cord ? 

9. What part of a pound of silver are 3 ounces ? What 
must I give for 3 ounces of silver at $ 12 a pound ? 

10. What part of a day are 3 hours ? 4 hours ? 20 hours ? 

For the Slate. 

11. What part of a bushel, in common and in decimal 
fractions, are 3 pecks, 4 quarts, 1 pint ? 

operation in common fractions, operation in decimals. . 
2)1 pt. 2)1.0 pt. 

8 ) 4£ qts. 8 ) 4.5 qts. 


4 ) P ks * 4)3.5 625 pks. 

of a bushel. Ans. .8 9 0 6 2 5 ot a bushel, 

in common fractions. Ans. in decimal fractions. 

Explanation. We first find what part of a quart, or of 2 
pints, the 1 pint makes ; then what part of a peck, or of 8 
quarts, the 4 quarts and part of a quart make ; and finally, 
what part of a bushel, or of 4 pecks, the 3 pecks and part 
of a peck make. See Fractions, lesson 61. We divide, 
in fact, just as we do to change pints to quarts, quarts to 
pecks, and pecks to bushels. Pursue a similar course in like 
cases. 


Note. The whole of the operation m common fractions is not shown, 
since 4s and 3 ij are changed to improper fractions before dividing them. 
See Fractions, lesson 72, example 7. 

12. How many pecks, quarts, &c., are there in of a 
bushel, and in .890625 of a bushel ? 


Explain how example 11, lesson 102, is performed. 

How, in fact, do we divide ? What must we do in like cases ? 





116 


REDUCTION OF COMPOUND NUMBERS. 

OPERATIONS. 

57 
4 


6 4)2 2 8(3 pks. 

1 9 2 

- „ , 

3 6 remainder, ff of a pk. 

8 

6 4)2 8.8 ( 4 qts. 

2 5 6 

3 2 remainder, ff of a qt. 
2 


6 4)6 4( 1 pt. 
6 4 


.8 9 0 6 2 5 
4 


pks. 3.56 2500 
8 


qts. 4. 5 0 0 0 0 0 
2 


pt. 1.0 00000 


Ans. 3 pks. 4 qts. 1 pt. 

Explanation. We first find how many pecks there are 
in and .890625 of a bushel, or of 4 pecks ; then how 
many quarts there are in the fractions of a peck, or of 8 
quarts ; and finally, how many pints there are in the frac¬ 
tions of a quart, or of 2 pints. We multiply , in fact, just as 
we do to change bushels to pecks , pecks to quarts , and quarts 
to pints. Pursue a similar course in like cases. 

Lesson 103. 

For the Slate. 

1. A blacksmith sold 3 quarters 16 pounds of iron ; what 
part of a ton did he sell, and what was it worth at $ 60 a 
ton ? Ans. of a ton, and it was worth $2.68, nearly. 

2. How many pounds are there in .25 of a quarter ? 

Ans. 7. 

3. How many ounces, pennyweights, and grains are 
there in of a pound of silver ? Ans. 4 oz. 16 pwts. 

4. What part of an ounce of gold, in decimals, are 13 

pennyweights 3 grains ? Ans. .65625 of an ounce. 


Explain how example 12, lesson 102, is performed. 

How, in fact, do we multiply ? What must we do in like cases? 









REDUCTION OF COMPOUND NUMBERS. H7 

5. -jJg- of a yard is sometimes called a nail ; now what 
part of a yard are 3 quarters of a yard and 3 nails ? 

Ans. |£. 

6. What is the value of .1325 of a mile in smaller de¬ 
nominations ? Ans. 1 fur. 2 rods 6 ft. 7.2 in. 

7. yV of a square mile is how many acres, and how much 
is it worth at $ 10 an acre ? 

Ans. 40 A., and it is worth $400. 

8. What part of a square rod, in decimals, are 187 

square feet ? Ans. .68687, nearly. 

9. .8 of a cord of wood contains how many feet of wood; 
and what is it worth at $ 1 a foot ? 

Ans. 6.4 ft. of wood, and it is worth $ 6.40. 

10. What part of a ton of hewed timber are 44 cubic 

feet and 86.4 cubic inches ? Ans. .881. 

Lesson 104* 

1. If Sidney coal is worth $ 9 a chaldron, what is the 

value of 14 bushels 3 pecks ? Ans. $3.69, nearly. 

Explanation. First change 14 bushels 3 pecks to dec¬ 
imals of a chaldron. 

2. If a grocer sells filberts at 4 cents a quart, how much 

will he get for f of a bushel ? Ans. $.85, about. 

Explanation. First change § of a bushel to quarts and 
decimals of a quart. 

3. A man sells beer at 6| cents a quart ; how much at 
this rate will f of a hogshead be worth ? Ans. $ 10.80. 

4. What part of a gallon of milk, in common fractions, 

are 2 quarts 1 pint ? Ans. |. 

5. A man has 1 pipe 1 hogshead and 20 gallons of wine; 
how many hogsheads and decimals of a hogshead has he ? 

Ans. 3.3175, nearly. 

6. If molasses is worth $.33| a gallon, what is £ of a 

hogshead worth ? Ans. $ 15.75. 

7. If a clock ticks 172,800 times a day, how many times 
will it tick in 6 hours, 3 minutes, 4 seconds ? 

Ans. 43,568 times. 

8. How many hours are there in | of a day ? Ans. 4. 

9. What part of the circumference of a circle are 7 de¬ 
grees and 12 minutes ? Ans. .02. 

10. How many degrees is | of the circumference of a 

circle ? Ans. 40° 


118 


REDUCTION OF COMPOUND NUMBERS. 


Lesson 105. 

1. 2 tons 3 hundred weight 1 quarter of cheese sold for 

8250 ; what did 1 ton bring ? Ans. 8 115.61, nearly. 

Explanation. What quantity in tons and decimals of a 
ton sold for 8 250 ? 

2. If 8 pennyweights 8 grains of silver make 1 silver 

spoon, how many of such spoons can be made from 5 
ounces of silver ? Ans. 12. 

3. A man travelled 10 miles 2 furlongs 20 rods in 3.3 

hours; how many minutes and seconds was he travelling 1 
furlong ? Ans. 2 min. 24 sec. 

4. A man bought 1 square mile and 360 acres of land 
for 8 1,000 ; how much did it cost him a square mile ? 

Ans. 8 640. 

5. I bought 2 tons 12 ^cubic feet of hewed timber for 
8 21 ; how much did I give a ton ? Ans. 89.33^. 

6. If 4 bushels 3 pecks of potatoes weigh 3324 pounds, 

how many pounds will 1 bushel weigh ? Ans. 70. 

7. 6 gallons 2 quarts 1 pint of milk were sold for 8 1.59; 

what was the price of a quart ? Ans. 6 cents. 

8. A vessel sailed 10 leagues in 3 hours 20 minutes ; 

how far did she sail in 1 hour ? Ans. 3 lea. 

9. A man rode 4 miles in 20 minutes ; how far would 

he go in 1 hour at this rate ? Ans. 12 miles. 

10. A vessel on the equator sailing west, passed over 39 

degrees 45 minutes 12 seconds of the circumference of the 
earth in 21 days ; how long was she in sailing over 1 de¬ 
gree ? Ans. 12 h. 40 min. 41 sec., about. 

Lesson 106. 

1. What part of a pound is TT jVtj- of a ton ? Ans. 

2. What part of a pound is .0001 of a ton ? Ans. .224. 

Explanation. Proceed the same as if you were asked 

how many pounds were in of a ton, and in .0001 of a 
ton ? 

3. What part of a ton is 4 of a pound ? Ans. 

Explanation. Proceed the same as you would to find 

what part of a ton 1 pound is. 

4. What part of a foot in decimals is ^ of an inch ? or 

which is the same thing, change £ of an inch to the decimal 
of a foot. Ans. .041667, nearly. 


REDUCTION OF COMPOUND NUMBERS. 


119 


5. Change .01 of a foot to the decimal of an inch. 

Ans. .12 of an in. 

6. A man sold of an acre of land for $ 12 ; at this 

rate how much is a square rod worth ? Ans. $ 24. 

7. Change .75 of 1 quarter of an acre to the decimal of 

an acre. Ans. .1875 of an A. 

8. What part of 1 foot of wood are .2 of a cord ; and 
what is 1 foot of wood worth if .2 of a cord be worth $ 1.60 ? 

Ans. .2 of a C. is 1.6 ft., and 1 ft. is worth $ 1. 

9. What part of a cubic foot, in common fractions, is £ 

of a cubic inch ? Ans. TT J ¥T . 

10. Change ^ of a bushel to the decimal of a pint. 

Ans. 1.28 pt. 

11. Change £ of a peck to the decimal of a bushel. 

Ans. .05 of a bu 

12. What part of a year of 365 days is ^ of a day ? 

Ans. 


Lesson 107. 

1. If 1 bu. of wheat purchase 2 bu. 3 pks. of potatoes, 

how many bu. of potatoes will 5 bu. 1 pk. of wheat pur¬ 
chase ? Ans. 14 bu. If pk. 

Explanation. First change 2 bu. 3 pks., and 5 bu. 1 pk. 
to bushels and decimals. 

2. A silversmith gave 1 oz. of silver for 2 cwt. of hay ; 

how much hay could he have obtained for 5 oz. 15 pwts. 
of silver ? Ans. 11 cwt. 2 qrs. 

3. If I give 5 lbs. 8 oz. of sugar for 1 ft. of wood, how 
much sugar must I give for 4 C. 7 ft. of wood ? 

Ans. 214.5 lbs. 

4. A vessel sailed 3 lea. in 1 hour ; how far would she 

sail in 2 d. 7 h. at the same rate ? Ans. 165 lea. 

5. What number of sq. rods are there in a field 16 rods 
12 ft. long, and 12 rods 5 ft. wide ? Ans. 205.796, about. 

6. How many sq. ft. are there in a mat 7 ft. 3 in. long, 

and 3 ft. 8 in. wide ? Ans. 26.58, about. 

Note. When feet and inches are to be changed to the same denom¬ 
ination, it is usually better to change them to feet and decimals, than to 
inches. It is easy to change feet and inches to feet and decimals, if we 
understand the table in lesson 92, Federal Money, for there are as many 


What is said of changing feet and inches to the same denomination? 
Why is it easy to change feet and inches to feet and decimals. 



120 


REDUCTION OF COMPOUND NUMBERS. 


hundredths of a foot in an inch as there are cents or hundredths of a 
dollar in a New England sixpence. Thus sixpence is $.08£, and 1 inch 
.08| of a foot. 1 shilling is $ .16§, and 2 inches .16§ of a foot. 1 shilling 
and sixpence, or 3 sixpences, are $ .25, and 3 inches .25 of a foot, &c. 

Engineers and some surveyors of wood and lumber, employ measures 
divided into feet and decimals; that is, divided into feet, tenths and 
hundredths of a foot. 

7. There is a cistern of water which is 13 ft. 4 in. long, 

6 ft. 9 in. wide, and 4 ft. 6 in. deep; how many cubic ft. of 
water are there in it ? Ans. 405. 

8. How many cords of wood are there in a pile 25 ft. 6 
in. long, 4 ft. wide, and 8 ft. 11 in. high. 

Ans. 7 C. 13.5 cubic ft. 

9. What part of 1 C. of wood is there in a pile 8 ft. long, 

4 ft. wide, and 3 ft. 3 in. high ? Ans. .8125. 

10. How many acres are there in a tract of land 1 mile 

264 ft. long, and i of a mile wide ? Ans. 336. 

Explanation. How many feet are there in a mile ? How 
long is the tract in miles and decimals ? 


Lesson 108. 

1. If 3 cwt. 2 qrs. of bread last a family 1 year, how 

long, in years and days, will 1 T. 2 cwt. I qr. 16.8 lbs. 
last them ? Ans. 6 yrs. 146 d. 

2. A farmer exchanged 2 T. 4 cwt. of hay for 2 hhds. 

17 gals, of molasses ; how many gallons of molasses could 
he have obtained for 1 T. of hay ? Ans. 65. 

3. A horse ran 1 mile in 3 min. 15 sec. ; how far could 
he run in 1 h. 20 m. 36 sec. at the same rate ? 

Ans. 24 m. 6 fur. 16 rods. 

4. How many rings, each weighing 2 pwts. 4 grs. can 
be made from J lb. 5 oz. 8 pwts. 20 grs. of gold ? Ans. 161. 

5. A piece of ground in the shape of an oblong square, 

containing 1 A. 3 qrs. 30 sq. rods, is 15 rods 8 ft. 3 in 
wide ; how long is it ? . Ans. 20 rods 

6. There is a door 3 ft. 3 in. wide, the surface of which 

is 23 sq. ft. 108 sq. in.; what is its height in feet and 
inches ? Ans. 7 ft. 3.7 in., nearly. 

Explanation. First get its height in feet and decimals. 


Give some examples. 

What measures do engineers and some surveyors of wood and lum¬ 
ber employ 



REDUCTION OF COMPOUND NUMBERS. 


121 


7. How high in feet and inches must you pile a load of 

wood which is 8 ft. long and 3 ft. 9 in. wide, so that it may 
contain a cord ? Ans. 4 ft. 3.2 in. 

Explanation. The number of cubic feet in a cord, divided 
by the surface of the bottom of the load, will evidently give 
the height. See Division, lesson 51. 

8. If a box contain 27 cubic ft., and the surface of one 

end be 4 sq. ft., what is its length ? Ans. 6 ft. 9 in. 

9. A cart containing 30 cubic ft., is 2 ft. 3 in. high; what 
is the surface of the bottom ? Ans. 13 sq. ft. 48 sq. in. 

10. The sur ace of the end of a stick of hewed timber, 

containing 1 T. 12 cubic ft., is 2^ sq. ft. ; what is its 
length ? Ans. 27 ft. 6§ in. 

Lesson 109. 

Compound numbers can be used in every respect as sim¬ 
ple numbers, if they are first changed to the same denomi¬ 
nation. 

1. A trader sold to three men the following quantities of 
wine. To the first 2 gals. 1 qt.; to the second 3 qts. 1 pt.; 
and to the third 2 qts. ; how much did he sell all of them ? 

Ans. 3 gals. 2 qts. 1 pt. 

Explanation. Before adding, we change all the quanti¬ 
ties to pints, or to quarts and decimals of a quart, or to gal¬ 
lons and decimals of a gallon. After adding, we change 
the sum to any denomination we please. 

2. Add 15 bu. 2 pks. 5 qts., 7 bu. 1 pk. 3 qts., and 23 

bu. 2 pks. Ans. 46 bu. 2 pks. 

3. If you are allowed to play 4 h. 20 min., how long can 
you remain after having played 2 h. 40 min. ? 

Ans. 1 h. 40 min. 

4. A man took 1 C. 7 ft. of wood from a heap that con¬ 
tained 6 C. 3 ft. ; how many cords were left ? 

Ans. 4 C. 4 ft. 

5. A milkman left 2 gals. 3 qts. 1 pt. of milk at a board¬ 

ing house every morning for 6 days ; what quantity did he 
leave in that time ? Ans. 17 gals. 1 qt. 

6. If you have 3 loads of hay, each weighing 18 cwt. 3 
qrs. 20 lbs., how many tons have you ? 

Ans. 2 T. 16 cwt. 3 qrs. 4 lbs. 


When can compound numbers be used in every respect as simple 
numbers ? 


11 



122 ADDITION OF COMPOUND NUMBERS. 

7. Multiply 33 ft. 7 in. by 4. Ans. 134 ft. 4 in. 

8. Divide 15 lbs. 9 oz. by 6. Ans. 2 lbs. 9 oz. 8 dr. 

9. 5 men owned 1 hhd. 1 bl. 16 gals, of wine in equal 
shares ; how much was each one’s portion ? 

Ans. 22 gals. 3.2 gi. 

10. How many square feet are there in 3 carpets, each 
of which contains 186 sq. ft. 66 sq. in. ? Ans. 559.375. 

It will be well to recollect that compound numbers can 
be used in this way, but we can add and subtract in these 
numbers, and can often multiply and divide them without 
the trouble of changing them to the same denomination as 
we shall now proceed to show. 

ADDITION OF COMPOUND NUMBERS. 

Lesson 110. 

To be performed in the mind. 

1. A man bought at different times the following quanti¬ 
ties of cinnamon ; 3 oz., 5 oz., and 9 oz. ; how many 
ounces did he buy ? How many pounds, &c. ? 

2. How many pecks of corn will you have after harvest¬ 
ing 2 pks., 3 pks., and 5 pks. ? How many bushels, &c., 
will you have ? How many bushels will you have after 
harvesting 1 bushel 3 pecks, and 3 bushels 2 pecks ? 4 
bushels 1 peck, 3 pecks, and 2 pecks ? 

3. A laborer worked for a merchant 4 h. 30 min.*at one 
time, 5 h. at another, and 12 h. 30. min. at another ; how 
many hours did he work ? 

4. What is the sum of 4 gals. 2 qts. 1 pt., 1 qt. 2 pts., 
and 3 pts. 

5. How many fathoms are there in 2 ft., 4 ft., 5 ft., and 
5 ft. ? 

For the Slate. 

6. A farmer sold at different times the following quanti¬ 
ties of butter ; 3 qrs. 12 lbs. 4 oz., 9 lbs. 14 oz., and 1 qr. 
15 oz.; what was the whole quantity he sold ? 


What else is said of compound numbers? 



ADDITION OF COMPOUND NUMBERS. 


123 


Explanation. We first place 
oz. under oz., lbs. under lbs., &c. 
Adding up the oz. we find there 
are 33, or 2 lbs., 1 oz.; we place 
the 1 oz. under the column of 
oz., and add the 2 lbs. with the 
lbs. There not being pounds 
Ans. 1 cwt. 0 qrs. 23 lbs. 1 oz. enough to make 1 qr., we write 
down the whole number, and then adding up the qrs., we 
get 4 qrs. or 1 cwt.; so we put 0 in the place of qrs. and 1 
in the place of cwts. 

Therefore, to add compound numbers, 

Write the numbers so that the quantities in the same de¬ 
nomination may stand directly under each other. Add the 
quantities in the lowest denomination first ; change the sum to 
the next higher denomination, and the result carry, and add 
with the next higher denomination, having first put down the 
remainder, and so on. 

The work is proved as in simple numbers. 

Note. Each example should now be proved. 


OPERATION, 
cwt. qrs. lbs. oz. 

3 12 4 

9 14 

1 0 15 


1 0 23 1 


Numbers to add. 

(7.) 

lb § 5 3 

5 11 5 2 

2 1 
27 0 0 0 


grs. 

16 

16 

18 


Numbers to add. 
( 8 .) 

chal. bu. pks. 

4 16 3 
12 0 3 

2 3 1 

6 18 1 


Numbers to add. 

(9.) 

miles, fur. rods. ft. 
2 0 27 0 

1 0 18 3 

36 10 

5 


10. A farmer sold four loads of hay, weighing as fol¬ 

lows ; the first 1 T., the second 16 cwt. 2 qrs. 18 lbs., the 
third 1 T. 2 cwt. 10 lbs., and the fourth 18 cwt.; what was 
the weight of the whole ? Ans. 3 T. 16 cwt. 3 qrs. 

11. If I buy the following quantities of oats, 12 bu. 3 pks. 
5 qts., 2 bu. 2 pks. 2 qts. 1 pt., 3 bu. 7 qts., 6 qts., and 1 
pk. 4 qts., what is the whole amount purchased ? 

Ans. 19 bu. 1 pk. 1 pt. 

12. Add the following quantities of silver ; 5 lbs. 11 oz. 
7 pwts. 3 grs., 6 lbs., 2 lbs. 15 grs., and 20 grs. 

Ans. 13 lbs. 11 oz. 8 pwts. 14 grs. 

Explain how example 6, lesson 110, is performed. 

How do we add compound numbers ? 

How is the work proved ? 







124 


ADDITION OF COMPOUND NUMBERS. 


Lesson 111. 

For the Slate. 

1 . How much beer is there in the following quantities ; 

2 hhds. 1 bl., 1 hhd. 2 bis. 6 gals., and 1 bl. 3 gals.? 

Ans. 5 hhds. 1 bl. 1 fir. 

2 . The first day a ship left port, she sailed 25 lea. 2,520 
fath., the second day, 13 lea. 180 fath., the third day, 17 
lea. 1,820 fath., and the fourth day, 3 lea. 2,260 fath.; how 
far did she sail during the four days ? 

Ans. 60 lea. 1,500 fath. 
Explanation. How many feet are there in a mile ? In a 
league ? How many fathoms then are there in a league ? 

3. How much land is therein three pieces, the first of 
which contains 17 A. 3 qrs. 12 sq. rods, the second 25 A., 
the third, 4 A. 6 sq. rods, and the fourth, 25 sq. rods ? 

Ans. 47 A. 3 sq. rods. 

4. How many cords of wood are there in £ of a C., 
2 § C., 3 C. 12 ft., 1 C. 10 ft., and 13 ft. ? 

Ans. 11 C. 4^ ft. 

Explanation. First change £ of a C. to feet, and 2 § C. 
to cords and feet. 

5. A grocer sold, at different times, the following quan¬ 
tities of molasses ; 4 hhds. 24 gals., 2 hhds. 2 bis. 4 gals. 

3 qts., and 24 hhds. ; what was the whole quantity sold ? 

Ans. 30 hhds. 2 bis. 28 gals. 3 qts. 

6 . Add 24 yrs. 55 d. 17 h., 2 yrs. 4\ d., and 5 d. 2 h. 

Ans. 26 yrs. 65 d. 1 h. 

7. Add 120 04' 13", 5o 12 ' 55", and 02' 07". 

Ans. 170 19 / i 5 '/ # 

8 . There is a bin of wheat, 8 ft. square on the bottom, 
and 4.5 ft. high, and another bin containing 17 bu. 4 pks. 

6 qts. ; how much wheat is there in both bins ? 

Ans. .249 bu. 2 pks. 3 qts. 1 pt., about. 
Explanation. First find how many cubic inches of wheat 
there are in the first bin, and recollect that there are 
2150.4 cubic inches in a bushel. 

9. A man has 4 pieces of rope ; the first is 2 yds. 2 ft. 

7 in. long, the second 24 yds., the third 7^- yds., and the 

fourth 3 yds. 1 ft. 11 in. long; what is the length of the four 
pieces ? Ans. 38 yds. 

10 . A man has a farm of 120 A. 3 qrs. 21 sq. rods, and 

another piece of land 27 rods long, and 17 rods wide; how 
much land has he ? Ans. 123 A. 3 qrs. 


SUBTRACTION OF COMPOUND NUMBERS. 


125 


SUBTRACTION OF COMPOUND NUMBERS. 


Lesson 112. 


To he performed in the mind. 

1 . If a grocer has 12 lbs. 9 oz. of butter, and sells 5 lbs. 
4 oz., how much will he have left ? 

2. A man having a stick 2 ft. 3 in. long, cut 1 ft. 2 in. 
off from it ; how long was the piece left ? 

3. If 1 gal. 3 qts. of wine leak out of a cask that con¬ 
tained 3 gals., what quantity will be left ? What quantity 
will be left if 3 qts. 1. pt. leak out of the cask ? 

4. A vessel employed 4 w. 1 d. in a voyage from Boston 
to Liverpool, having stopped 1 w. 2 d. at Cork ; how long 
was she at sea ? 

5. I sold 10 sq. rods from a piece of land containing 2 
quarters ; what quantity of land was left in the piece ? 


For the Slate. 

6 . A coal dealer having 112 chal. 4 bu. 3 pks. of coal, 
sold 38 chal. 17 bu. 2 pks. ; what quantity was there left ? 
operation. Explanation. We first take the 

2 pks. from 3 pks.; then, being un¬ 
able to take the 17 bu. from 4 bu., 
we add 1 chal., or 36 bu. to 4 bu., 

- and take 17 from 40. As we have 

73 23 1 added 1 chal. to the greater num¬ 

ber, to balance it, we now add 1 
the smaller number, and take 39 


chal. 

112 

38 


bu. 

4 

17 


pks. 

3 

2 


Ans. 73 chal. 23 bu. 1 pk. 

chal. to 38 chal. in 
from 112 . 


Therefore, to subtract one compound number from an¬ 
other, 

Subtract the quantity in the lowest denomination of the 
smaller number from that above, and set down the remainder, 
and so on. When the quantity we are to subtract from is the 
smallest, add as many to it as make one of the next higher 
denomination, subtract, and then carry 1 to the next higher de¬ 
nomination of the smaller number. 

The work is proved as in simple numbers. 

Note. Each example should now be proved. 

Explain how example 6, lesson 112, is performed. 

How do we subtract one compound number from another ? 

How is the work proved ? 




126 


SUBTRACTION OF COMPOUND NUMBERS. 


(7.) (8.) (9.) 

T.cwt. qrs. lbs. ft) §59 grs. h. m * n * sec * 

From 4 13 2 19 From 3 10 4 1 12 From 17 21 12 

take 2 17 3 20 take 1 0 0 1 15 take 10 17 5 


10 . A trader having 215 bu. of beans, sold 38 bu. 3 pks. 
4 qts.; what quantity had he left ? Ans. 176 bu. 4 qts. 

11 . A grocer bought 4 bis. 8 gals. 2 qts. of beer, but lost 

I bl. 7 gals. 3 qts. by leakage; how much did he have left ? 

Ans. 3 bis. 3 qts. 

12. Subtract 7 lbs. 10 oz. 15 pwts. 17 grs., from 9 lbs. 

II oz. 17 pwts. 21 grs. Ans. 2 lbs. 1 oz. 2 pwts. 4 grs. 

Lesson 113. 

For the Slate . 

1 . A merchant having f of a T. of rice, sold 3 qrs. 17 
lbs.; how much had he left ? Ans. 12 cwt. 1 qr. 20 £ lbs 

Explanation. First change f of a T. to smaller de¬ 
nominations. 

2 . A landholder who owned 3 sq. miles, sold 712 A. 2 
qrs. 35 sq. fods; how much did he retain ? 

Ans. 1,207 A. 1 qr. 5 sq. rods. 

3. There are two cities 98 miles 5 furlongs 3 rods apart; 
how far is a man travelling between these cities from one 
of them if he is 12 miles 6 T ^ furlongs from the other ? 

Ans. 85 miles 6 furlongs 39 rods. 

4. A man bought some hewed timber for 7 T. 12 cubic 
ft., but on measuring it, the quantity fell short 30 cubic ft. 
1,200 cubic in.; how much timber was there ? 

Ans. 6 T. 31 cubic ft. 528 cubic in. 

5. Take l pt. from 2 bu. Ans. 1 bu. 3 pks. 7 qts. 1 pt. 

6 . Take 10 cubic in. from 3 T. round timber, when ^ is 

allowed for waste. Ans. 2 T. 39 cubic ft. 1,718 cubic in. 

7. Take 2 in. from 3 yds. Ans. 2 yds. 2 ft. 10 in. 

8 . Take 1 pt. from 17 gals. 2 gi. 

Ans. 16 gals. 3 qts. 1 pt. 2 gi. 

9. Take 4 grs. from 1 lb. Ans. 11 oz. 19 pwts. 20 grs. 

10 . Take 1 d. from 2 w. 10 sec. Ans. 1 w. 6 d. 10 sec. 

11 . A wine dealer having 2 tier. 2 bis. and 12 gals, of 

Madeira wine, sold 100 gals. ; how many gallons had he 
left ? Ans. 59. 

12 . From 7 w. 3 d. 16 h. 5 min. 28 sec., take 1 w. 4 d. 
17 h. 16 min. 39 sec. Ans. 5 w. 5 d. 22 h. 48 min. 49 sec. 





MULTIPLICATION OF COMPOUND NUMBERS. ]27 

13. A certain city is 71° 18' 45" west of Greenwich, and 

another 89° 36' ; how many degrees, minutes, &c. is one 
of these cities west of the other ? Ans. 18° 17' 15". 

14. A farmer made 185£ bis. of cider, and sold 123 bis. 
5 gals. 1 pt.; how much had he left ? 

Ans. 62 bis. 18 gals. 2 qts. 

15. A goldsmith having 2 lbs. 4 oz. 15 grs. of gold, used 

1 lb. 17 pwts. in manufacturing some articles ; what quan 
tity had he left ? Ans. 1 lb. 3 oz. 3 pwts. 15 grs 

MULTIPLICATION OF COMPOUND NUMBERS. 

Lesson 114. 

To be performed in the mind. 

1 . There is a basket which holds 1 bu. 1 pk. 2 qts.; how 
much corn is there in a heap that contains 3 such baskets 
full ? 4 such baskets full ? 5 ? 6 ? 

2 . A man bought of a farmer 6 kegs full of cider ; what 
quantity, in gallons, &c., did he get if the kegs held 5 qts. 
each ? What if they held 6 qts. each ? 7 qts. each ? 1 qt. 
1 pt. each ? 

3. If you study arithmetic 1 h. 15 min. each day, how 
much time will you occupy in that study in 2 days ? In 3 
days ? In 4 days ? In 5 days ? 

4. What is the product of 4 yds. 2 ft. by 2 ? By 3 ? .By 

5 ? By 8 ? 

For the Slate. 

5. A butcher sold 4 loads of beef, each of which con¬ 
tained 6 cwt. 2 qrs. 21 lbs.; what was the whole quantity sold ? 

operation. j Explanation. The 21 lbs. multipli- 

T. cwt. qrs. lbs. ed by 4, give 84 lbs., or 3 qrs., 0 lbs.; 
6 2 21 go W e put down 0 in the lbs. place, 

4 and add the 3 qrs. to the product of the 

-T “ 2 qrs. by 4. These 11 qrs. are equal 

* ^ ^ ^ to 2 cwt. 3 qrs.; so we put down the 
Ans. 1 T. 6 cwt. 3 qrs. 3 q rs>> and add the 2 cwt. to the pro¬ 
duct of the 6 cwt. by 4. These 26 cwt. are equal to 1 T. 

6 cwt., which we put down. 

Therefore, to multiply a compound number by a whole 
number, 


Explain how example 5, lesson 114, is performed. 





128 MULTIPLICATION OF COMPOUND NUMBERS. 

Multiply each of the denominations, beginning with the 
lowest , and carry as in addition of compound numbers. 

The work is proved by dividing the product by the mul¬ 
tiplier; if the quotient be equal to the multiplicand, the 
work will generally be right. See Division, lesson 51. 

Note. Each example should now be proved. 

6 . Multiply 4 chal. 17 bu. 2 pks. by 8. 

7. Multiply 2° 15' by 90. 

8 . A wagon is loaded with 12 bales of cotton, each of 

which weighs 4 cwt. 27 lbs. 5 oz. ; what is the weight of 
the load ? Ans. 2 T. 10 cwt. 3 qrs. 19 lbs. 12 oz. 

Explanation. Each bale is 4 cwt. 0 qrs. 27 lbs. 5 oz. 

9. What is the weight of 28 ingots of silver, each of 
which contains 1 lb. 17 grs. ? Ans. 28 lbs. 19 pwts. 20 grs. 

10. How much are 5 times 21b 71 23 17 grs. ? 

Ans. 131b 33 19 5 grs. 

11 . Multiply 2 miles 275 rods by 37. 

Ans. 105 miles 255 rods. 

Explanation. Find how many rods there are in a mile. 

12. A speculator divided some land into 6 house lots, 
each of which contained 1 A. 4 sq. rods 120 sq. ft.; what 
quantity of land was there in all of the lots ? 

Ans. 6 A. 26 sq. rods 175£ sq. ft. 

13. What is the amount of 2 times 5 C. 5 ft. 178 cu¬ 
bic in. ? Ans. 11 C. 2 ft. 356 cubic in. 

14. Multiply 2 hhds. 1 kil. 1 fir. by 3. 

Ans. 7 hhds. 1 kil. 1 fir. 

15. What is the product of 5 qts. 1 pt. 2 gi. by 125 ? 

Ans. 179 gals. 2 qts. 1 pt. 2 gi. 

16. What is the product of 1 d. 18 h. 10 sec. by 5 ? 

Ans. 1 w. Id. 18 h. 50 sec. 

DIVISION OF COMPOUND NUMBERS. 

Lesson 115. 

To be performed in the mind. 

1. 3 soldiers obtained 3 lbs. 9 oz. of silver from the plun¬ 
der of a city; if they divide it equally, what will each one’s 
share be ? What would each one’s share have been had 
they obtained 4 lbs. ? 4 lbs. 6 oz. ? 4 lbs. 3 oz. ? 


How do we multiply a compound number bv a whole number ? 
How is the work proved ? 



DIVISION OF COMPOUND NUMBERS. J29 

2. If 2 men have equal shares in 3 bushels of potatoes, 
what is the portion of each in bushels and pecks ? 

3. What is the quotient of 6 w. 3 d. 12 h. divided by 3 ? 
Of 1 d. 12 h. divided by 4 ? 

4. If you have 2 qts. 1 pt. of chestnuts to divide among 
2 boys, what quantity must you give to each ? 

For the Slate. 


Way of proceeding when the divisor is not more than 10. 

5. A farmer wishes to carry 34 bu. 2 pks. 5 qts. of salt 
in 3 equal loads; how much must he put in each load ? 


bu. 

3)34 


OPERATION, 
pks. qts. pts. 

2 5 


Explanation. Dividing 34 bu. 
by 3, we get 11 bu. and 1 bu. 
over. Changing this 1 bu. to 
pks., adding them to 2 pks. and 
dividing the sum, 6 pks., by 3, 
we get 2 pks. Dividing the 5 
qts. by 3, we get 1 qt. and 2 qts. over. Changing the 2 
qts. to pts., and dividing, we get 1£ pt. 


11 2 1 1 £ 
Ans. 11 bu. 2 pks. 1 qt. 1$ pt. 


Way of proceeding when the divisor is more than 10. 


6 . What is the quotient of 1 w. 3 d. 54 min. divided by 18 ? 

OPERATION. 


w. d. 

18)1 3 

7 

7(0 w. 
Add 3 days. 

10(0 d. 
24 

40 

20 

240(13 h. 
18 


h. min. w. d. h. min. 

0 54 ( 0 0 13 23 Ans. 


6 brought up for 
60 room. 

360 

Add 54 min. 

414(23 min. 

36 


54 

54 


Explanation. 
We proceed as 
before, but for 
the sake of ease 
write down all 
of the work. 
As 18 is not 
contained in the 
1 week, nor in 
the 10 days, we 
get no weeks 
nor days in the 
answer. 


60 

54 


6 carried up. 

Explain how example 5, lesson 115, is performed. 
Explain how example 6 , lesson 115, is performed. 




130 


DIVISION OF COMPOUND NUMBERS- 


Therefore, to divide a compound number by a whole 
number, 

Divide the highest denomination first, and the remainder , 
if any, change to the next lower denomination , add it to the 
number in that denomination, and divide the result as before , 
and so on. 

The work is proved as in simple numbers. 

Note. Each example should now be proved. 

7. Divide 5 lbs. 11 oz. 12 grs. by 6. 

8. Divide 7§ 6 3 19 by 3. 

9. If I buy 1 T. 5 cwt. 2 qrs. of hay for 8 25, how much 
can I get for $ 1 at the same rate ? Ans. 1 cwt. 2^1bs. 

10. Divide 4 miles 55 rods by 12. Ans. 111^ rods. 

11. Divide 17§ A. by 15, and get the answer in acres, 
quarters, and square rods. 

Ans. 1 A. 28.444 sq. rods, about. 

12. If 11 C. 5 ft. 10 cubic ft. of wood cost 860, what 
quantity can I get for 8 l at the same rate ? 

Ans. 1 ft. 8.9§ cubic in. 

13. Divide 4 chal. 0 bu. 3 pks. by 7. Ans. 20 bu. 2-f- pks. 

14. If 2 fir. of beer be divided equally among 50 men, 

what quantity will each receive ? Ans. 1 qt. 3§§ gi., 

15. Divide 5 pipes of wine into 50 equal parts. 

Ans. 12 gals. 2 qts. 3£ gi. 

16. What is the quotient of 14° divided by 27 ? 

Ans. 31' 6§". 

MULTIPLICATION AND DIVISION OF COMPOUND NUMBERS BY 
FRACTIONS AND MIXED NUMBERS. 

Lesson 116. 

Note. If there are decimals in the numbers by which we are to mul¬ 
tiply or divide, they must first be changed to common fractions, unless 
we "proceed as directed in lesson 109, Reduction of Compound Numbers. 

For the Slate. 

1. If you are to have § of a load of hay that weighs 1 T 
2 qrs. 19 lbs., what quantity will that be ? 

Ans. 13 cwt 3 qrs. 3£ lbs. 


How do we divide a compound number by a whole number ? 

How is the work proved ? 

If there are decimals in the numbers by which we are to multiply or 
divide, what is to be done with them ? 




PROMISCUOUS QUESTIONS. 


131 


Explanation. Observe the rule in Fractions, lesson 70, 
and the two preceding rules for multiplying and dividing 
compound numbers. 

2 . A man gave $ 8 £ for 1 chal. 7 bu. 3 pks. of Sidney 

coal ; how much could he have obtained for $ 1 , at the 
same rate ? Ans. 5 bu. 

Explanation. Observe the rule in Fractions, lesson 73, 
and the two preceding rules for multiplying and dividing 
compound numbers. 

3. Multiply 6 lbs. 2 oz. 10 grs. by 

Ans. 9 oz. 5 pwts. 1 ^ gr. 

4. Multiply 10 S 23 by 4£. Ans. 31b 8 § 33 19. 

5. Divide 10 sq. rods 2.25 sq. ft. by . 6 . 

Ans. 16 sq. rods 185.25 sq. ft. 

6 . If you buy 4 C. 9 ft. of wood for $24|-, how much 
can you buy for $ 1 , at the same rate ? 

Ans. 1 ft. 10f£ cubic ft., or 1 ft. 10 £ cubic ft., about. 

7. A trader sold ^ of a bin of wheat for $ 12.50 ; the 
bin, on measurement, was found to contain 18 bu. 3 pks.; 
how much did he sell for $ 12.50, and how much for $ 1 ? 
Ans. 7 bu. 3 pks. 2 qts. for $ 12.50, and 2 pks. 4 qts. for $ 1 . 

8 . How much is 4f times 2 hhds. 1 bl. 7 gals, of wine ? 

Ans. 12 hhds. 25.375 gals. 

9. A laborer being 8 h. 45 min. doing a piece of work, 

another man agreed to perform a similar job in £ of the 
time ; how long was that ? Ans. 7 h. 17 min. 30 sec. 

10. A portion of the circumference of a circle, contain¬ 

ing 11° 34' 06", is to be divided into equal parts ; how 
large will each portion be ? Ans. 2° 06' 12". 


PROMISCUOUS QUESTIONS 

IN 

FEDERAL MONEY AND COMPOUND NUMBERS. 

Lesson 117. 

To be performed in the mind. 

1 . A piece of land is divided into two house lots, one of 
which is 25 ft. 8 in. wide, and the other 30 ft. 6 in. wide ; 
what is the width of the whole piece ? 



132 


PROMISCUOUS QUESTIONS. 


2 . If the piece had been divided into two lots of equal 
widths, how wide would each have been ? 

3. If I buy 3 C. 6 ft. of pine wood for 610, how much 
do I get for 6 1 ? What do I give a foot for it ? 

4. A trader had 16 gals. 2 qts. of vinegar in a cask; how 
much was there left after selling 3 qts. ? I gal. 2 qta. ? 3 
gals. 3 qts. ? 

5. A girl in Boston bought a silver pencil case for 75 
cents, a sheet of drawing paper for 1 shilling, and four 
quills for ninepence ; she gave a 2 -dollar bill in payment ; 
what sum did she receive back ? 

6 . What quantity of silver is there in 4 silver pitchers, 
each of which weighs 1 lb. 5 oz. ? 

7. A man in Albany paid 3 shillings and sixpence for a 
cane, and 6 3£ for an umbrella ; what sum did he give for 
both ? 

8 . How many square yards are there in a piece of ground 
15 ft. square ? 

9. What must I give for 16 yds. of cloth at 6 2 ^- a yard ? 

10 . How many hundred weight are there in £ of a ton 
of anthracite coal ? How many pounds ? 

11 . How will you contrive to pay ~a man 70 cents, in 
Pittsburg, if you have in your purse, £ of a dollar, \ of a 
dollar, 2 levies, 3 tips, and 4 cents ? How near can you 
pay him the exact sum ? 

12 . A man bought | of a load of hay containing 1 T. 
4 cwt. ; what quantity did he obtain ? 

Lesson 118 . 

For the Slate. 

1 . A man paid 6 .90 for ^ of a quantity of hay, esti¬ 

mated to be 2 T. ; what price a hundred weight did he 
give ? Ans. 6 .56£. 

2 . If a boatman in New York charge you 6 shillings 

for rowing you over the Hudson River and back again, 
how many cents must you pay him ? Ans. 75 cts. 

3. A man owns one farm containing 115 A. 2 qrs. 13 
sq. rods ; another containing 37 A. 30 sq. rods, and a third 
containing 18 A. 3 qrs. ; how much land has he ? 

Ans. 171 A. 2 qrs. 3 sq. rode. 

4. A trader sold 2 bu. 1 pk. of beans for 6 5.94 ; what 

price did he get a peck ? Ans. 6 . 66 . 


PROMISCUOUS QUESTIONS. 


133 


5. If you retail 1 quintal of cod fish at 4 cents a pound, 
2 bis. of beef at 10 cents a pound, and 1 bl. of flour at 3| 
cents a pound, what sum will you get for the whole ? 

Ans. $ 51.83. 

6 . What part of a pint, wine measure, is of a 

hogshead ? Ans. i. 

7. A young man is 19 years 86 days old, and his sister 
17 years 119 days old; what is the difference of their ages ? 

Ans. 1 yr. 332 d. 

8 . A man boarded in Northampton, Massachusetts, 12 

weeks, at 15 shillings a week; what sum in Federal money 
must he pay his landlord ? Ans. $ 30. 

9. A man sold 1 A. 2 qrs. 8 sq. rods of land at 8 cents 
a square foot ; what sum did it bring him ? 

Ans. $5,401.44. 

10 . What part of a cord of wood, in common fractions, 

are 48 cubic ft. ? ‘ Ans. |. 


Lesson 119. 

1 . The distance between two headlands on the coast of 
Maine, was measured, and found to be 6,530 fathoms ; 
what is this distance expressed in miles and rods ? 

Ans. 7 m. 134-^ rods. 

2. If a man pump 27 gals. 3 qts. 1 pt. of water in 1 
minute, how much can he pump in 9 h. 4 min. 30 sec. ? 

Ans. 15,177 gals. 3 qts. 1 pt. 2 gi. 

3. What must I pay for 2 T. 7 cwt. 3 qrs. of hay, at the 

rate of $ 17 a ton ? Ans. $40.58£. 

4. How many square feet are there in a garden 5 rods 
12 ft. 6 in. long, and 2 rods 8 ft. wide ? Ans. 3,895 sq. ft. 

5. How many hours and minutes are there in of a 

day ? Ans. 12 h. 48 min. 

6 . 6 silver tea spoons weigh 2 oz. 9 pwts. 12 grs. ; what 

is the weight of each ? Ans. 8 pwts. 6 grs. 

7. 2 T. 3 cwt. 2 qrs. of American bar iron was sold for 

$ 130.50 ; what price a ton did it bring ? Ans. $60. 

8 . How many pieces of iron, each weighing 1 lb. 4 oz., 
can be cut off from a bar weighing 2 cwt. 1 qr. 8 lbs. ? 

Ans. 203 pieces. 

9 . If you sell 3£ tubs of butter, each of which weighs 
56 lbs. 13 oz. what is the whole quantity disposed of? 

Ans. 198 lbs. 13£ oz. 


134 


PERCENTAGE. 


10. There is a block of wood 4 ft. long, and 2 ft. 6 in. 
high, containing 12.5 cubic ft. ; how wide is it ? 

Ans. 1 ft. 3 in. 

Explanation . See Division, lesson 51. 


PERCENTAGE. 


Lesson 120. 


Per cent, is a contraction of the Latin per centum , and 
signifies per hundred. 

1 per cent, of any number is of it, or .01 of it ; 
2 per cent, is .02 of it, 3 per cent, is .03 of it, 25 per cent, 
is .25 of it, 120 per cent, is 1.20 of it, and 100 per cent, of 
any number is just equal to it, &c. Also, £ per cent., or 
£ of 1 per cent, of any number, is £ of .01, or .005 of it ; £ 
per cent, is .0025 of it, and 2^ per cent, is .025 of it, &c. 

Any per cent, of a number, then, is so many hundredths 
of that number. 


Write on your slate answers to the following questions. 


What is 
What is 
What is 
What is 
What is 
What is 
What is 
What is 
What is 
What is 
What is 
What is 


6 per cent, of any number ? 
£ per cent, of any number ? 
5£ per cent, of any number ? 
200 per cent, of any number ? 
17 per cent, of any number ? 
2^ per cent, of any number ? 
525 per cent, of any number ? 
33£ per cent, of any number ? 
10.12 per cent, of any number ? 
20 per cent, of any number ? 

per cent, of any number ? 
100 per cent, of any number ? 


What is per cent, a contraction of, and what does it signify ? 

What is 1 per cent, of any number ? 2 per cent. ? 3 per cent. ? 25 
per cent. ? 120 per cent. ? 100 per cent. ? £ per cent. ? \ per cent. P 
2 £ per cent. ? , 

What is any per cent, of a number ? 




PERCENTAGE. 


135 


What per cent, of any number, that 

is, how many hun- 

dredths of any number will 

.005 

of that number be ? 

will 

£ 

of that number be ? 

will 

.185 

of that number be ? 

will, 

.12125 

of that number be ? 

will 

£ 

of that number be ? 

will 

.0025 

of that number be ? 

will 

.44 

of that number be ? 

will 

b 

of that number be ? 

will 

2£ times 

that number be ? 

will 

5 times 

that number be ? 

will 

12£ times 

that number be ? 

will 

7£ times 

that number be ? 


The number of which we are to get a certain per cent, 
is called the principal , and the per cent., considered as 
hundredths, is called the rate. 


Lesson 121 . 

1. A trader intrusted a man with $ 225, 4 per cent, of 
which he was directed to invest in books, and the rest in 
West India goods ; what sum must he lay out in books ? 

OPERATION. 

$225 
.0 4 


$ 9.0 0 Ans. 

2. A man let a house in Baltimore for a rent of $ 950 a 

year, with a condition that 8^- per cent, of the rent should 
be expended in repairs on the house ; how many dollars a 
year must be laid out in repairs ? Ans. $ 80.75. 

3. A merchant having $2,507.75 in a bank, drew out 
20 per cent, of it ; how much was that ? Ans. $501.55. 

4. What is 435 per cent, of $ 1,000 ? Ans. $4,350. 

5. A merchant had 3 bales of cotton, the first of which 
contained 4 cwt. 1 qr. 27 lbs., the second 5 cwt. 7 lbs., and 
the third 3 cwt. 3 qrs. 8 lbs.; they were so much damaged 
during a fire that 18 per cent, of the cotton was destroy¬ 
ed ; what quantity was that ? 

Ans. 2 cwt. 1 qr. 17 lbs. 10.24 oz. 


What is called the principal ? The rate ? 




136 


COMMISSION. 


Since we multiply the principal by the rate to get the 
amount of the per cent., 

We must divide the amount of the per cent, by the rate to 
get the principal , and by the principal to get the rate. 

6. A merchant having $ 4,000 deposited in a bank, drew 
out a certain per cent, of it, the amount of which was 
$ 20 ; what per cent, of the deposit was that ? 

Ans. £ per cent. 

7. The owner of an iron mine allowed some laborers to 

work it for one year, on condition of paying him 10 per 
cent, of the proceeds ; at the end of the year they paid 
him $ 250 ; what were the proceeds ? Ans. $ 2,500. 

8. 17 bu. 2 pks. are 7£ per cent, of a certain quantity ; 

what is that quantity ? Ans. 233 bu. 1} pk. 

9. If I pay a man $8, and have $ 1,992 left, what per 
cent, of my money do I pay him ? Ans. of 1 per cent. 

Explanation. $8 added to $ 1,992 is the sum which I 
had, and $ 8 is the amount of the per cent. I pay. 

10. If a man paid me $ 114, which was 5 per cent, less 
than what he owed me, how much did he owe me ? 

Ans. $ 120. 

Explanation. He paid 95 per cent, of what was due. 

11. If a trader sends you 75 yards of black broadcloth, 

20 per cent, more than you ordered, what quantity did you 
order ? Ans. 62£ yds 


COMMISSION. 

Lesson 122 . 

Commission is a reward paid to an agent, factor, broker, 
or correspondent, of so much per cent, on the amount of 
the purchases or sales made by him. 

1. What amount of commission must I pay my factor for 
selling $ 3,525.16§ worth of rice, at 2£ per cent. ? 

Ans. $ 79.32. 


How do we get the principal from the amount of the per cent, and 
rate, and the rate from the amount of the per cent, and principal ? 
Why ? 

What is commission ? 




STOCKS. 


137 


2. If I buy 375 chal. 35 bu. 3 pks. of Newcastle coal at 

$ 12 a chal., and receive 2 per cent, commission, what will 
be the amount of my commission ? Ans. $ 90.24. 

3. The agent of a landholder sold 117 A. 2 qrs. 4 sq. 
rods of land, at $ 18 an acre, and charged 1^ per cent, 
commission ; what sum must be paid to the landholder 
after deducting the amount of the commission ? 

Ans. $2,087.24. 

4. A broker charged $ 35 for assisting me in purchasing 

a farm for $ 5,000 ; what per cent, commission or broker¬ 
age did he demand ? Ans. ^ of 1 per cent. 

5. The expenses of selling a quantity of copper, at 2 per 

cent, commission, were $253.42; what was the sum it sold 
for? , Ans. $ 12,671. 

6. My correspondent sold 2,525 bu. of potatoes for me, 
at a commission of 3 per cent., and sent me $979.70 as 
the balance due after deducting the amount of his commis¬ 
sion ; how much did he sell them all for, and how much 
did he get a bushel ? 

Ans. he sold all for $1,010, and got 40 cents a bu. 

7. If a man sends you $2,050 to lay out in goods for 
him, and directs you to retain 2£ per cent, on the amount 
of purchases for commission, what sum must you retain ? 

Ans. $ 50. 


STOCKS. 

Lesson 123 . 

Stock is the name of the funds of government, and of 
the capital of banks, insurance offices, factories, canals, 
railroads, and like companies. It is owned in shares. 

When a share of any stock sells at its original cost, it is 
said to be at par; when it sells at more than its original 
cost, it is said to be above par , and at so much per cent. 
advance; when it sells at less than its original cost, it is 
said to be below par , and at so much per cent, discount. 


What is stock ? How is it owned ? 

When is stock said to be at par ? Above par ? Below par ? In ad¬ 
vance ? At a discount ? 

12 * 




138 


BANKRUPTCY. 


1. A man bought 23 shares of the Utica and Schenecta¬ 
dy railroad stock at 21A per cent, advance, the par value 
being $ 100 a share ; what did he give for them ? 

Ans. $2,794.50. 

2. If I buy several shares of bank stock at $54 a share, 

the par value being $ 60, at what per cent, discount do I 
obtain them ? Ans. 10 per cent. 

3. A merchant bought a share of the stock of the Locks 
and Canals Company in Lowell at 270 per cent, advance ; 
what did it cost, the par value being $500 ? Ans. $ 1,850. 

4. A broker bought 5 shares of the Massachusetts Bank 

for me, at 1 per cent, advance ; what did they cost me, the 
par value being $ 250 a share, and the broker charging f 
per cent, commission ? Ans. $ 1,271.97. 

5. If you buy canal stock at 40 per cent, discount, and 
give $ 30 a share, what is the par value ? 

Ans. $ 50 a share. 

6. What is the par value of a share of factory stock 
which sold for $ 560 at 12 per cent, advance ? 

Ans. $ 500 a share. 


BANKRUPTCY. 

It sometimes happens that a merchant, through miscal¬ 
culation or misfortune, becomes unable to pay the full 
amount of his debts, in which case he is called a bankrupt , 
and his property is distributed among his creditors in pro¬ 
portion to what is due to each. 

7. A man failed for $3,528, having property to the 
amount of $2,963.52 ; what per cent, of his debts can he 
pay ; that is, what per cent, of $3,528 is $ 2,963.52 ? 

Ans. 84 per cent. 

8. How much will he pay on a debt of $ 100 ? Ans. $ 84. 

9. Samuel Jackson failed, owing as follows ; to John 
Smith $ 1,800, to Charles Brown $ 8,350, to John Williams 
$2,511.16§, and to James Thompson $ 5,000 ; if he has 
property to the amount of $ 10,596.70, what per cent, of 
his debts can he pay, and what can he pay each of his 


What sometimes happens to a merchant? In that case, what is he 
called, and what is done with his property ? 




LOSS AND GAIN. 


139 


creditors ? Ans. he can pay 60 per cent, of his debts ; to 
John Smith $1,080, Charles Brown $5,010, John Wil¬ 
liams $ 1,506.70, and James Thompson $ 3,000. 

10. A bankrupt who pays 33^ per cent, of his debts, 
owes me $ 180 ; how much can I get of it ? Ans. $60. 

11. A bankrupt who has property to the amount of 
$ 16,525, owes A. $ 10,000, B. $ 5,000, C $3,828, and D. 
$ 6,500 ; if the above are all his debts, what per cent, of 
them can he pay, and what sum will each of his creditors 
receive ? Ans. he can pay 65-^/j^y per cent, of his debts, 
nearly; to A. $6,524.40, B. $3,262.20, C. $2,497.54, 
and D. $ 4,240.86. 


LOSS AND GAIN. 

Lesson 124. 

1. A merchant bought a quantity of molasses for $ 3,500; 
for what sum must he sell it so as to gain 10 per cent.? 

Ans. for $ 3,850. 

2. If I buy 5 pieces of broadcloth, containing 27 yards 

each, at $4 a yard, and sell the whole for $583.20, what 
per cent, do I gain ? Ans. 8 per cent. 

Explanation. The first cost is the principal, and the 
gain is the amount of the per cent. 

3. Having bought a quantity of wine for $873.25, I 

lost so much by leakage and other accidents, that I was 
content to sell it for 18 per cent, less than the first cost; 
what did I get for it ? Ans. $ 716.06J-. 

4. A merchant bought $ 2,000 worth of goods, and 
marked the prices 33£ per cent, higher than the cost, but 
in selling the goods he took 25 per cent, less than was 
marked ; what sum did he gain or lose ? 

Ans. he sold the goods at first cost. 

5. What per cent, would he have gained or lost had he 
taken from the price marked 25 per cent, of the first cost ? 

Ans. he would have gained 8£ per cent. 

6. A man bought 35 T. 18 cwt. of hay for $600 ; at 
what price a ton must he sell it so as to gain 12£ per cent. ? 

Ans. $ 18.80. 

7. If your agent buys one thousand bushels cf corn for 



140 


DRAFT AND TARE. 


$ 750, and charges 2 per cent, commission, at what per 
cent, in advance of the cost, which includes commission, 
must you sell the corn to gain $ 350 ? 

Ans. 45J per cent., about. 

8. A merchant bought a quantity of pork at $25 a bar¬ 

rel, and put the price 30 per cent, more than the cost; 
what sum must he take off from the price of each barrel 
to gain 10 per cent.? Ans. $5. 

9. Sold 700 barrels of flour for $4,550, gaining 4 per 
cent, on it ; what did it cost me a barrel ? Ans. $6.25. 

10. A merchant bought 65 hogsheads of molasses in 

Matanzas, at $21.50 a hogshead ; after v paying $ 50.59 for 
duties, $79.33 for freight, $39 for insurance to Portland, 
$ 12 for truckage, and $4.50 for sundry other expenses, 
he finally sold it at $ 30 a hogshead ; what per cent, did 
he gain ? Ans. 23^^ per cent., about. 

11. A man sold a ship for $ 11,500, gaining 15 per cent, 
on what she cost him ; what per cent, would he have 
gained by selling the ship for $ 10,800 ? Ans. 8 per cent. 


DRAFT AND TARE. 

Lesson 125. 

Draft is a small allowance on the weight of an article so 
that the quantity may hold out when retailed. 

The following table shows the allowances usually made 
for draft. 

Note. It is not necessary to commit it to memory. 

The allowance for draft on a parcel weighing 

112 lbs. is 1 lb. 

above 112 lbs. and not over 224 lbs. is 2 lbs. 

above 224 lbs. and not over 336 lbs. is 3 lbs. 

above 336 lbs. and not over 1,120 lbs. is 4 lbs. 

above 1,120 lbs. and not over 2,016 lbs. is 7 lbs 

above 2,016 lbs. is 9 lbs. 

Tare is an allowance for the weight of the box, cask, 


What is draft ? What is tare ? 




DRAFT AND TARE. 


141 


bag, 8cc., containing the goods. It is reckoned on what is 
left after the draft has been deducted. Draft and tare are 
the only allowances now made by merchants. 

The following table shows the allowances usually made 
for tare on several important articles of merchandise. 

Note. It is not necessary to commit it to memory. 

The allowance for tare on 

Almonds in bags, is. 3 per cent. 

Alum in casks,.12 per cent. 

Bristles from Cronstadt,.12 per cent. 

Bristles from Archangel,.14 per cent. 

Beef, jerked, in hogsheads,.112 lbs. a hhd. 

Beef, jerked, in drums,.70 lbs. a drum. 

Cordage in mats,.If per cent. 

Camphor, crude, in tubs,.35 per cent. 

Candles in boxes,. 8 per cent. 

Cinnamon in chests,.16 lbs. a chest. 

Cinnamon in mats,. 8 per cent. 

Cloves in casks,.15 per cent. 

Cocoa^in bags,. 1 per cent. 

Cocoa in casks,.10 per cent. 

Cocoa in serons,.10 per cent. 

Chocolate in boxes,...10 per cent. 

Coffee in bags from the West Indies,.... 2 per cent. 
Coffee in grass bags from the East Indies, 2 lbs. a bag. 

Coffee in bales,. 3 per cent. 

Coffee in casks,.12 per cent. 

Cotton in bales,.2 per cent. 

Cotton in serons,. 6 per cent. 

Currants in casks,.12 per cent. 

Cheese in hampers or baskets,.10 per cent. 

Cheese in boxes,.20 per cent. 

Copperas in casks,.12 per cent. 

Figs in boxes of 60 lbs.,. 9 lbs. a box. 

Figs in half boxes of 30 lbs.,.5f lbs. a box. 

Figs in quarter boxes of 15 lbs.,.3f lbs. a box. 

Figs in drums,.10 per cent. 

Glue, from Russia, in bales,.5 lbs. a bale. 

Glauber salts in casks,. 8 per cent. 


How is tare reckoned ? What are the only allowances now made by 
merchants. 

































142 DRAFT AND TARE. 

Indigo in bags or mats,.3 per cent. 

Indigo in serons,.lO^per cent. 

Indigo in barrels,..12 per cent. 

Indigo in other casks,.15 per cent. 

Indigo in cases,.20 per cent. 

Mace in casks or kegs,.33 per cent. 

Nails in casks,. 8 per cent. 

Ochre, French, in casks,.12 per cent. 

Pepper in bags,. 2 per cent. 

Pepper in bales,. 5 per cent. 

Pepper in casks,.12 per cent. 

Pimento in bags,.3 per cent. 

Pimento in bales, . 5 per cent. 

Pimento in casks,...16 per cent. 

Prunes in boxes.7 lbs. a box. 

Raisins, Malaga, in boxes, .. 6 lbs. a box. 

Raisins, Malaga, in jars,.5 lbs. a jar. 

Raisins* Malaga, in casks,.12 lbs. a cask. 

Raisins, Smyrna, ..12 per cent. 

Sugar, Java, in willow baskets,.60 lbs. a basket. 

Sugar in bags or mats,. 5 per cent. 

Sugar in casks,.12 per cent. 

Sugar in boxes,.15 per cent. 

Sugar in canisters,.35 per cent. 

Sugar candy in baskets*.5 per cent. 

Sugar candy in boxes, . ..10 per cent. 

Soap in boxes,.10 per cent. 

Soap, Marseilles, in boxes,.12 per cent. 

Shot in casks,. 3 per cent. 

Steel in bundles,.2 lbs. a bundle. 

Steel in cases,...60 lbs. a case. 

Tea, bohea, in chests,. .......70 lbs. a chest. 

Tea, bohea, in half chests,. .36 lbs. a piece. 

Tea, bohea, in quarter chests,.20 lbs. apiece. 

Twine in casks,.12 per cent. 

Twine in bales,.3 per cent. 

Tallow in serons,.10 per cent. 

Tallow in casks,..12 per cent. 

Wool, Smyrna, in bales,.10 lbs. a bale. 

Wool, Hamburg, in bales,. 3 per cent. 

Wool, South American, in bales,.15 lbs. a bale. 


The weight of the envelopes, of articles not named in the 
table is estimated, and the tare fixed accordingly. 


How is the tare fixed on articles not named in the table. 












































DRAFT AND TARE. 


143 

Jfole. Tare is always expressed in whole pounds ; less than half a 
pound is omitted ; more than half a pound is reckoned a pound. 

The whole weight of any parcel of goods, including 
the box, cask, bag, &c., containing the goods, is called the 
gross weight. The weight of any parcel of goods after 
the draft and tare have been deducted, is called the neat 
weight. 

Leakage . After the exact quantity of liquid in a cask 
is found, by gauging, it is usual to deduct 2 per cent, for 
leakage. 

1. What is the neat weight of 20 casks of Malaga 

raisins, the gross weight of each being 128 lbs., the draft 
and tare as in the tables ? Ans. 2,280 lbs. 

2. A merchant bought 3 bales of cotton; the gross weight 

of the first was 5 cwt. 23 lbs.; of the second, 4 cwt. 3 qrs.; 
and of the third, 3 cwt. 3 qrs. 26 lbs.; how much cotton 
did he get ? Ans. 13 cwt. 2 qrs. 6 lbs. 

3. If I buy 10 bags of West India coffee, the gross 

weight of each of which is 1 cwt., how much coffee do I 
obtain ? Ans. 1,088 lbs. 

4. The gross weight of 17 bags of pepper is 17 cwt.; 
how much is the neat weight ? Ans. 16 cwt. 2 qrs.l lb. 

5. What is the neat weight, in pounds, of 27 hhds. of 

sugar, the average gross weight of each being 7 cwt. 1 qr. 
6 lbs.? Ans 19,341 lbs. 

6. What amount of leakage must be allowed on three 
hhds. of rum, which being gauged were found to contain 
the following quantities ; the first, 68 gals. 3 qts., the sec¬ 
ond 60 gals., and the third, 65 gals. 5 qts.? 

Ans. 3.88 gals., say 4 gals. 

7. What is the draft and tare on 4 casks of nails, the 
average gross weight of each being 217 lbs.? 

Ans. the draft is 8 lbs., and the tare 69 lbs. 

8. The gross weight of a firkin of butter is 50 lbs., and 

the tare is reckoned 6 lbs. ; what per cent of the whole is 
butter ? Ans. 88 per cent. 


In what is tare always expressed ? What is done with less than half 
a pound ? With more than half a pound ? 

What is called gross weight? Neat weight? 

What is said of leakage ? 



144 


DUTIES. 


DUTIES. 

Lesson 126. 

Duties are taxes on many kinds of goods imported into 
the United States from foreign countries, and are collected 
by Custom House Officers, appointed by government. 

The duty on some kinds of goods is a certain per cent. 
of their legal value, and is called an ad valorem duty. The 
legal value of goods subject to an ad valorem duty, is their 
original cost, which includes all the original charges, ex¬ 
cept insurance. 

The duty on other kinds of goods is so much a ton, hun¬ 
dred weight, pound, gallon, yard, &.c. and is called a spe¬ 
cific duty; allowances for draft, tare, and leakage are made 
before calculating it. 

Debenture is a remission of the duty on foreign goods re¬ 
exported. 

Drawback is the amount of such a remission. 

Dounty is a reward paid to the exporter of certain do¬ 
mestic goods. 

Note. When the duties on any goods are less than $50, the re¬ 
exporter is not entitled to debenture. 

1. What is the duty on a quantity of woollen goods 
that cost in England $ 10,275, at 44 per cent, ad valorem ? 

Ans. $4,521. 

2. If I import 18 boxes of chocolate, the gross weight 

of each being 1 cwt., what sum must I pay for duty, at 4 
cents a pound ? Ans. $71.92. 

3. What will the bounty 6n 1,637 bis. of American 
shad exported to Messina, amount to at 20 cents a bl. ? 

Ans. $ 327,40. 

4. What is the duty on 7 pipes of Port wine, contain¬ 
ing on an average 130 gallons apiece, at 15 cents a gallon ? 

Ans. $ 133.80. 


What are duties, and how collected ? 

What is called an ad valorem duty ? What is the legal value of goods 
subject to an ad valorem duty ? What does the original cost include ? 

What is called a specific duty ? What allowances are made before 
calculating it ? 

What is debenture ? Drawback P Bounty ? 

When is the re-exporter not entitled to debenture ? 



SIMPLE INTEREST. 


145 


Explanation. Remember the leakage. 

5. What will the drawback amount to on a quantity of 

silk goods that cost $ 8,237, the duty being 10 per cent, ad 
valorem ? Ans. $ 823.70. 

6. In a catalogue of goods with their prices, &c., called 

an invoice of goods, from Matanzas, I find 16 boxes of 
brown sugar, the average gross weight of each of which is 
7 cwt. 2 qrs. 6 lbs., the tare being stated at 17 per cent. ; 
if the invoice tare be allowed, what will the duty be at 2£ 
cents a pound ? Ans. $ 279.55. 

7. What is the duty on a cargo of iron imported from 
Newport, in Wales, containing 275 T., at $ 30 a ton, no 
allowance being made for draft or tare ? Ans. $ 8,250. 

8. How much drawback am I entitled to on exporting 7 
puncheons of rum, containing, according to measurement, 
800 gallons, the duty being 42 cents a gallon ? 

, Ans. $ 329.28, 

9. A merchant of Boston, exported 230 hhds. of New 

England rum, averaging 112 gals, apiece, to Smyrna; 
what sum must he receive for bounty, the law allowing 
him 4 cents a gal. ? Ans. $ 1,009.80, 


INTEREST, 

OH THE REWARD PAID FOR THE USE OF MONEY, 


SIMPLE INTEREST. 

Lesson 127. 

Simple interest is a reward of so much a year for the 
money lent. A certain per cent, of the money is generally 
paid for its use one year. 

The money lent is called the principal; the per cent, 
paid for the use of the money one year, is called the rate; 
and the principal and interest added together are called 
the amount. 

What is simple interest ? What is generally paid for the use of 
money one year ? 

What is called the principal ? Rate ? Amount ? 

13 





146 


SIMPLE INTEREST. 


Note 1. In speaking of interest, when we say 1. per cent., 2 per cent., 
3 per cent., &c., we mean 1 per cent., 2 per cent., 3 per cent., &c., 
a year , unless some other time is stated. 

Note 2. The law in most of the states allows the creditor to receive 
only 6 per cent.; but there is nothing to hinder an agreement at a 
lower rate. 

1. What is the interest of $125 for 1 yesn’, at 6 per 
cent. ? 

OPERATION. 

$12 5 principal. 

. 0 6 rate. 


$7. 5 0 interest. Ans. 

2. If you lend a man $2,750.25 for 1 year, at 5 per cent., 
what interest must he pay you at the end of that time ? 

Ans. $137.51. 

3. How much will the principal and interest both amount 
to ; that is, what will be the amount ? Ans. $2,887.76. 

4. What is the interest of $1,723.33 for 1 year, at 5j 

per cent. ? Ans. $99.09, 

5. A man borrowed $1,800 for 3 years, at 6 per cent. ; 

what will be the interest for that time ? Ans. $324. 

Explanation. How many times the interest for 1 year 
will the interest for 3 years be ? 

6. What is the interest of $1,000, at 4£ per cent., for 12 

years ? Ans. $585. 

7. If I lend $12,230 for 5 years, at 6 per cent., what 
amount must I be paid at the end of that time ? 

Ans. $15,899. 

8. What is the greatest amount of interest a man can 
get lawfully in 6 years for $1,250, in the state of New 
York, where the legal rate is 7 per cent. ? Ans. $525. 

9. What is the amount of the principal and interest on 
$1,605.05 for 64 years, at 6f per cent. ? 

Ans. $2,300.57. 


In speaking of interest, when we say 1 per cent., 2 per cent., 3 per 
cent, &c., what do we mean ? 

What is the highest per cent, the law allows the creditor to receive 
in most of the states ? Can an agreement be made at a lower rate ? 




SIMPLE INTEREST. 


147 


10. What sum must I give for the use of $ 250 during 7 

years, at 6 per cent. ? Ans. Si05. 

11. What will $ 16.25 amount to in 2£ years, at 6 per 

cent. ? Ans. $18.69. 


Lesson 128, 

1. What is the interest of $100.33 from January 1st, 
1835, to January 16th, 1837, at 6 per cent. ? 

OPERATION. 

$1 0 0.33 principal, $6.02 nearly, inter- 

. 0 6 rate, 1 5 est for 1 yr. 

6.0198 interest for 1 yr. 3010 

years, time. 6 0 2 

_ _<jj» 

1 2.03 9 6 3 6 5) 9 0 3 0 (.2 5 nearly, 

. 2 5 interest for 15 days. 7 3 0 interest 

- - for 15 d. 

$12.29 interest for 2 years 1730 

15 days. Ans. 18 2 5 

Explanation. We find the interest for one year, and as 
15 days are of a year, we multiply the interest of 1 
year by 2-g^ years, the time. We have cast the interest 
to the nearest cent. 

From the preceding we derive the following 

RULE FOR SIMPLE INTEREST. 

Multiply the principal by the rate , and the product by the 
time , in years and 365Ms of a year. 

2. A merchant borrowed $2,000 the 20th of January, 
1821, at 6 per cent., and paid the debt the 12th of Febru¬ 
ary, 1822 ; how much was the interest for that time ? 

Ans. $127.56. 

The table on the following page will enable us to find 
part of a year in days, with great ease. 


Explain how example 1, lesson 128, is performed. 
What is the rule for Simple Interest ? 









148 


SIMPLE INTEREST. 


TABLE 


Showing the number of days between any two months in 
one year. 


From |Jan. |Feb. |Mar. |April|May |June|July | Aug.|Sept-|Oct. |Nov.|Dec. 

To Jan. |365|334|306|275|245|214 184|l53|l22| 92 | 61 j 31 

Feb. 

| 3l|365 337 306|276|245|215|l84|l53|l23 92| 62 

March 

| 59| 28|365|334|304|273|243 212|l811151 |l20| 90 

April 

|' 901 59| 3l|365|335|304|274 243|212|l32|l51 121 

May 

[l20| 89| 61 30|365|334|304273|242|212 18l|l51 

June 

11511120| 92 611 3l|365j335|304|273|243|212 182 

July 

|l8l|l50|l22| 9 1 1 611 30 365|334|303|273|242|212 

Aug. 

|212| 181 153 122| 92 611 311365|334|304|273 243 

Sept. 

|243|212|l84|l53 123 92| 62 3l|365335304|274 

Oct. 

|273|242|214|l83|l53|l22| 921 6l| 30 365|334 304 

Nov. 

|304 273|245l214|l84|l53|123| 92| 611 3l|365|335 

Dec. 

|334|303|275244|214|l83|l53|l22| 911 6l| 30|365 


The use of this table is explained by the following ex¬ 
amples ; 

How many days are there between Feb. 5th and June 
12th ? Look in the column under Feb., and opposite 
June you find 120, the number of days between Feb. Sth 
and June 5th ; add the 7 days between June 5th and June 
12th, and you get 127 days, the answer. 

How many days are there between Sept. 20th and 
March 8th ? Look in the column under Sept., and oppo¬ 
site March you find 181, the number of days between 
Sept. 20th and March 20th ; subtract the 12 days between 
March 8th and March 20th, and you get 169 days, the 
answer. 

In leap years, if the end of Feb. be in the time, 1 day 
must be added to the number found in the table. 


How do you find the number of days between Feb. 5th and June 
12th by the table ? 

How do you find the number of days between Sept. 20th and March 
8th by the table ? 

How is the table used in leap years? 



















SIMPLE INTEREST. 


149 

Note. Let the scholar compute the time in each example of this 
lesson, and compare it with the time found by the table. 

3. A farmer gave his note July 17th, 1827, for $116.50 
at 7 per cent, interest, and the 4th of March, 1828, he paid 
principal and interest ; what sum was the interest ? 

Ans. $5.16. 

Explanation. How many days were there in February, 
1828 ? 

4. If a man gave his note May 7th, 1830, for $1,800, at 

6 per cent., what sum was necessary to discharge the debt 
June 21st, 1834 ? Ans. $2,245.32. 

5. A man borrowed $605.25 for 1 year and 20 days, at 

6£ per cent. ; what interest was due at the time of pay¬ 
ment ? Ans. $43.09. 

6. If a note was given for $1,000, at 7 per cent., to be 

paid in 2 years and 10 days, what interest was due at the 
end of that time ? Ans. $141.92. 

7. What sum will a note given September 1st, 1836, for 

$2,800, at 5f per cent., amount to the 1st of the next 
July ? Ans. $2,931.72. 

8. Cast the interest on a note for $71.30, at 6 percent., 
given December 18th, 1830, and paid August 7th, 1837. 

Ans. $28.39. 

9. What is the interest on $2,500.50 for 28 days, at 10 

percent. ? Ans. $19.18. 

10. What will be the amount of $5,000 for 211 days, at 

6 per cent. ? Ans. $5,173.42. 

11. Cast the interest on the following note, at 7 per 
cent., paid January 12th, 1827, 

$800. Buffalo* May 1st, 1819. 

For value received, I promise to pay John Howard, or 
order, eight hundred dollars, on demand, with interest. 

Edward Jones. 

Ans. $431.28. 

Lesson 129 . 

Another common hut incorrect method. 

To obtain the interest for part of a year many persons 

Take as many 12 ths of the interest for 1 year as there are 
months , and as many 30 ths of the interest for 1 month as 
there are odd days . 


How do many persons obtain the interest for part of a year? 

13 * 



150 


SIMPLE INTEREST. 


The time from the 1st of one month to the 1st of the 
next, is called a month, also from the 2d of one month to 
the 2d of the next, from the 3d of one month to the 3d of 
the next, and so on. 

Note. This method may be employed with propriety when the time 
in a note or agreement is expressed in months. It may often be 
abridged by considering 2 months as I of a year, 3 months as 1, 4 
months as »; 6 months as |; 8 months as and 9 months as | of a 
year. 

1. What is the interest by the preceding method on a 
note for $88, at 7 per cent., given July 15th, 1817, and 
paid December 28th, 1818 ? 

operation. $ 

$88 12) 6.16 ( .513 about, in. for 1 mo. 

.07 6 0 5 mo. 


6.16 interest for 1 yr. 16 2.565 in. for 5 mo. 

2.57 interest for 5 mo. 12 

.22 interest for 13 d. - 

- 40 

$8.95 in. for 1 yr. 5 mo. 36 

13 d. Ans. $ 

30 ) .513 ( .017 about, interest 
30 13 for 1 d. 


213 51 

210 17 

.221 in. for 13 d. 

Explanation. The time being 1 year 5 months and 13 
days, we first get 1 year’s interest, which divided by 12 
gives the interest for 1 month, and this multiplied by 5 
gives the interest for 5 months. The interest for 1 month 
divided by 30, gives 1 day’s interest, which multiplied by 
13 gives the interest for 13 days. Finally, adding the 
interest for 1 year 5 months and 13 days together, we get 
the whole interest. 

2. A man lent $ 850.25 for 8 months, at 5 per cent. ; 
what was the interest for that time ? Ans. $ 28.34. 


What is called a month? 

When may this method be employed with propriety ? How may it 
often be abridged ? 

Explain how example 1, lesson 129, is performed. 








SIMPLE INTEREST. 


151 


3. What is the interest on $75.50 for 4 months todays, 

at 8 per cent. ? Ans. $2.21. 

4. What interest was due on a note given March 22d, 

1831, for $1,738, at 6 percent., and paid June 6th, 1836, 
by the preceding rule ? Ans. $543.12^. 

5. A merchant borrowed $100 the 20th of May, 1835, 
at 2 per cent, a month, and paid principal and interest the 
4th of the next August ; what did the interest amount to ? 

Ans. $5. 

6. If I borrow $1,500, at 2£ per cent, a month, and pay 
the debt in 5 months and 18 days, what will be its amount ? 

Ans. $1,710. 

7. What is the interest on $2,000 for 90 days, at 6 per 

cent. ? Ans. $30. 

Explanation. When money is lent for 30, 60, or 90 
days, 30 days are called a month. 

8. If you lend $500 at 1 per cent, a month for 60 days, 

and are paid at the end of 70 days, what amount must you 
receive ? Ans. $511.66§. 

9. A man borrowed $150 for 30 days, at 10 per cent. ; 
what will be the amount at the end of that time ? 

Ans. $151.25. 

10. What was the interest by the preceding rule on a 

note for $18.25, at 8 per cent., given April 3d, 1830, and 
paid June 6th, 1832 ? Ans. $3.18. 


Lesson 130 . 

We have seen that the interest is obtained by multiply¬ 
ing the principal, rate, and time together ; 

Therefore, when we know any two of these three things , 
the principal, rate, and time, by dividing the interest by their 
product, we get the other. See Division, lesson 51. 

Note. The principal and interest added together, compose the 
amount; so the principal subtracted from the amount leaves the inter¬ 
est, and the interest subtracted from the amount leaves the principal. 

1. A man paid $5.35 for the use of $55.20 during 1 


When we know any two of these three things, the principal, rate, and 
time, how do we find the other ? 

What things compose the amount ? How then do you get the inter¬ 
est from the amount and principal ? The principal from the amount 
and interest ? 



152 


SIMPLE INTEREST. 


year 7 months and 12 days ; what rate per cent, did he 
give ? Ans. 6 per cent. 

Explanation. First change the days and months to frac¬ 
tions of a year. 

2. If I receive $163.12£ as the amount of $150 for 1 year 
91 days, what rate per cent, do I get ? Ans. 7 per cent. 

3. How longmust $2,515.31£remain at interest, at 6 per 
cent., so that it may amount to $2,615.37£ ? Ans. 242 d. 

4. I owe a man $17.50 ; how long must I let him use 
$87, at 6 per cent., so that the interest may pay the debt ? 

Ans. 3 yrs. 129 d, about. 

5. My agent paid me $37.25 as the interest due on 

money lent for 2 years 17 days, at 7 per cent. ; what was 
the sum lent ? Ans. $260.02. 

The rate , time, and amount given to find the principal. 

6. What sum will amount to $37.50 in 2£ years, at 6 
per cent. ? 

operation. $ 

$1 1.15 ) 37.50 ( 32.61 nearly. Ans. 

06 rate. 34 5 


.06 interest of $1 for 1 yr. 300 
2£ 230 


700 

690 


.15 interest of $1 for 2£ yrs. 100 
1 115 


$1.15 amount of $1 for 2^- yrs. 

Explanation. We find the amount of $1 for the rate 
and time; and divide the amount, $37.50, by it. It will evi¬ 
dently be contained in $37.50 as many times as there are 
dollars in the principal. 

Therefore, to obtain the principal when we know the 
rate, time, and amount, 

Divide the amount hy the amount ofi $ 1. 

7. What principal, at 5 per cent., will amount to $100 
in 1 year 105 days ? Ans. $93.95. 


12 

3 


Explain how example 6, lesson 130, is performed. 

How do we obtain the principal when we know the rate, time, and 
amount ? 







SIMPLE INTEREST. 


153 


8. What principal, at 7 per cent., will amount to $25.124 

in 4 years 3 months ? Ans. $19.36*. 

9. A certain sum lent at 6 per cent, produced $250 be¬ 

tween July 5th, 1830, and December 26th, 1831 ; what 
was that sum ? Ans. $229.65. 

10. What sum will amount to $10 in 1 year, at 6 per 

cent. ? Ans. $9.43. 


Lesson 131. 

To cast the interest on notes, bonds, fyc., when partial pay¬ 
ments have been made. 

The rule for this purpose adopted by the Supreme Court 
of the United States, and by the Courts in most of the 
States is as follows ; 

Cast the interest to the time when the money paid shall at 
least be equal to the interest, then discharge the interest from 
the money paid, subtract the excess, if any, from the princi¬ 
pal, and cast the interest on the new principal as before, and 
so on. 

1 . - 

$327. 

- Boston, June 4th, 1829. 

For value received, I promise to pay John Smith, or 
order, three hundred and twenty-seven dollars, on de¬ 
mand, with interest. George Brown. 

On this note were the following endorsements ; 

April 16th, 1831, received $100. 

July 4th, 1831, received $3. 

January 12th, 1832, received $18.25. 

October 1st, 1832, received $55.16. 

What was due January 1st, 1833 ? 

OPERATION. 

Payment April 16th, 1831,.$100. 

Subtract interest on $327 from June 4th, 1829 to 

April 16th, 1831,. 36,61 

Excess, . 63,39. 


Recite the rule generally employed in the United States to obtain 
the interest on notes, bonds, &c., when partial payments have been 
made. 









154 


SIMPLE INTEREST. 


From principal.$327.00. 

Subtract excess, . 63.39. 


Remainder,. 263.61. 

Payment July 4th, 183i,.$3.00. 


Interest on $263.61 from April J 6th, 1831, 
to July 4th, 1831, $3.42, which being 
more than the payment, we must cast 
the interest to the time when the payments 
amount to as much or more than the in¬ 
terest. 

Payment January 12th, 1832..18.25. 

Sum of the two last payments,.21.25. 

Subtract interest on $263.61 from April 
16th, 1831, to January 12th, 1832, .... 11.74. 


Excess,.. 9.51. 

From remainder subtract excess,.9.51 


Remainder,.. 254.10. 

Payment October 1st, 1832, .. 55.16. 

Subtract interest on $254.10 from January 
12th, 1832, to October 1st, 1832, .. 10.99. 


Excess,.44.17. 

From last remainder subtract excess,.44.17. 


Remainder,. 209.93. 

Add interest on $209.93 from October 1st, 1832, 

to January 1st, 1833,. 3.17. 


Ans. due January 1st, 1833,.$213.10. 


Natchez, June 12th, 1819. 

For value received, I promise to pay James Waldron, 
or order, five thousand two hundred and fifty-five dollars 
and fifty cents, in one year, with interest afterwards. 

Charles Laval 

On the back of this note were written the following re¬ 
ceipts ; 

August 2d, 1822, received $600. 

December 14th, 1822, received $1,000. 

March 1st, 1823, received $2,260.37. 

























SIMPLE INTEREST. 


155 


What was due July 1st, 1825, 6 per cent, being allowed 

Ans.$2,555.88. 

3. $625. Schenectady, December 1st, 1830. 

For value received, we, jointly and severally, promise 
to pay Peter Vanderheyden, or order, six hundred and 
twenty-five dollars, in four years from date, with interest 
till paid. Hermann Van Pelt. 

Walter Suydam. 

On this note were the following endorsements ; 

February 1st, 1831, received $180. 

January 2d, 1832, received $5.50. 

February 20th, 1832, received $2. 

March 15th, 1833, received $200. 

How much must the creditor receive when the note be¬ 
comes due ? Ans. $349.35. 

Explanation. Schenectady is in New York, where 7 
per cent, is the legal interest. 


4. $.210.14. Wheeling, August 13th, 1828. 

For value received, I promise to pay Alexander Stevens, 
or order, two hundred and ten dollars and fourteen cents, 
on demand, with interest. William Doyle. 

Witness, Samuel White. 

On this note were the following endorsements ; 

December 12th, 1828, received $3. 

May 2d, 1829, received $5. 

June 1st, 1829, received $180.25. 

What was due July 29th, 1829, reckoning ^6 per cent, 
interest ? Ans. $32.28. 

5. Suppose I give a man in Boston a bond for $600, 
dated June 1st, 1835, and drawing interest, by which I am 
required to pay $100 at the beginning of every month for 
5 months, and at the end of 6 months to pay the balance ; 
what will the balance be if I pay promptly ? 

Ans. $110.68. 

6. What is due July 1st, 1836, on a note given August 
3d, 1835, for $25.50, drawing 6 per cent, interest, on the 
back of which were the following endorsements ; 

January 13th, 1836, received $18.12. 

February 1st, 1836, received $2. Ans. $6.24 





156 


SIMPLE INTEREST. 


Lesson 132. 

Another common but incorrect method. 

To obtain what is due on a note or bond when partial 
payments have been made, many persons, 

Find the amount of the principal at the time of settlement , 
also the amount of each payment at the time of settlement , and 
then subtract the sum of the amounts of the payments from 
the amount of the principal. 

This method is easy but illegal, and is considered unjust 
towards the lender ; lor if you lend me $100 at 6 per cent., 
and I pay you the interest, or $6, at the end of every year, 
in 25 years the amounts of the payments will be more than 
the amount of the principal. 

Experiment, however, shows it may be employed with¬ 
out much error, when a settlement is made within a year 
of the commencement of interest on the note or bond. It 
may be used with propriety to find the balance due on an 
account at interest. 

Perform the following examples by this rule. 

1. Example 4, lesson 131. Ans. $32.08. 

2. Example 5, lesson 131. Ans. $110.50. 

3. Example 6, lesson 131. Ans. $6.22. 

4. What is the balance due on the following account, 

settled January 1st, 1835 ; each item drawing interest at 
6 per cent. ? v 


Samuel Jay’s account with David Sibley, 


Dr. 





Cr. 

1834. 



1834. 



March 1. 

To Goods 

$200 

June 2. 

By Cash 

$100 

May 5. 

“ Goods 

50 

Aug. 12. 

“ Cash 

85 

Sept. 8. 

“ Pork 

130 

Dec. 20. 

“ Cash 

125 


Ans. $78.77, due David Sibley. 


Explanation. The easiest course is to multiply each 
item on the Dr., or debtor side by the time in days, multi¬ 
ply the sum of the products by the rate, or .06, and divide 
this product by 365. The quotient is plainly the whole 

How do many persons obtain what is due on a note or bond, when 
partial payments have been made ? 

What is said of the ease, legality, and justice of this method ? 

When may it be employed without much error P When also may it 
be used ? 




SIMPLE INTEREST. 


157 


interest on the Dr. side. Proceed the same way to get 
the interest on the Cr., or creditor side. 

5. Find the sum due on the following account at inter¬ 
est, settled August 11th, 1838, 7 per cent, being reck¬ 
oned. 


R. Leach, his account current with J. Barr. 


Dr. 




1838. 


1837. 

Cr. 

Feb. 21. To Cash 

875 

Oct. 10. By Hay 

8250 

June 11. “ Cash 

200 

1838 


July 6. “ Cash 

175 

Jan. 18. “ Corn 

875 



Feb. 6. “ Oats 

60 



April 11. “ Potatoes 

100 


Ans. 851.04 due JL Leach. 

6. What is due on the following account at interest, 
settled July 1st, 1830, money being worth 5 per cent. ? 
John Marsh’s account current with James Colburn. 

Dr. 


1830. 


1830. 


Cr. 

Jan. 2. To Goods 

no 

Feb. 12. 

By Beef 

850 

June. 2. “ Goods 

100 

March 1. 

‘ ‘ Pork 

60 

1 


May 4. 

“ Hay 

20 


Ans. 89.27 due James Colburn. 

Note. The rule employed in the Courts of New Jersey, is but very 
little different from that given at the beginning of lesson 131. The 
rules employed in the Courts of Connecticut and Vermont, produce 
results a little different from this rule. 

The following is the rule established by the Supreme Court of the 
State of Connecticut in 1804, and should be studied by residents in that 
State. It may be omitted by others. 

CONNECTICUT RULE. 

Compute the interest to the time of the first payment; if that he one 
year or more from the time the interest commenced, add it to the principal , 
and deduct the 'payment from the sum total. If there he after payments 
made, compute the interest on the balance due, to the next payment, and 
then deduct the payment as above ; and in like manner from one payment 
to another till all the payments are absorbed; provided the time between 
one payment and another be one year or more. But if any payments be 
made before one year's interest hath accrued, then compute the interest on 
the principal sum due on the obligation for one year, add it to the pri.nci 
pal, and compute the interest on the sum paid, from the time it was paid 
up to the end of the year; add it to the sum paid, and deduct that sum 


What is said of the rule employed in the Courts of New Jersey ? Of 
the rules employed in the Courts of Connecticut and Vermont ? 

Recite the Connecticut Rule. 

14 





158 


COMPOUND INTEREST. 


from the principal and interest, added as above. However , if the year 
extends beyond the time of settlement, find the amount of the principal re¬ 
maining unpaid, up to the time of such settlement, also the amounts of the 
payment or payments up to the same time, and deduct their sum from the 
amount of the principal. 

If any payments be made of a less sum than the interest arisen at the 
time of such payment, no interest is to be computed, but only on the princi¬ 
pal sum for any period. 

Example 1, lesson 131, is performed by this rule, thus; 

Principal,.$327.00. 

Add interest from June 4th, 1829, to April 16th, 1831,. 36.61. 


363.61. 

Deduct first payment,.•. 100.00. 


263.61. 

Add interest of $263.61 for 1 year,.15.82. 


279.43. 

Deduct amount of second and third payments, at the end of 


this year, or April 16th, 1832,. 21,58. 

257.85. 

Add interest on $257.85 to the time of settlement,.11.02. 


268.87. 

Deduct amount of fourth payment at time of settlement,.55.99. 

Due January 1st, 1833,...$212.88. 


22 cents less than the answer found by the first method given. 

Explanation. We found the amount of the second and third pay¬ 
ments from the time the third payment was made ; for when the second 
payment was made, the interest on the principal sum amounted to more 
than the payment. 


COMPOUND INTEREST. 

Lesson 133 . 

Compound Interest is a reward paid for the principal, 
and also for the interest after it becomes due. 

To calculate compound interest, therefore, 

Make the amount at the time interest is due a new princi¬ 
pal, on which cast the interest to the time when interest is 
again due, and so on, and finally subtract the first principal 
from the last amount. 


What is compound interest ? 

How do we calculate compound interest ? 




















COMPOUND INTEREST. 


159 


1. Suppose I give a note, on demand, for $100, at 6 
per cent., and at the end of each year give a new note for 
the principal and interest; what will all the interest come 
to in 3 years ? 

OPERATION. 

$100 principal. 

.06 


6.00 

100 . 


106. amount, principal for the second year. 
.06 


6.36 interest for the second year. 
106 


112.36 amount, principal for the third year. 
.06 


6.7416 interest for the third year. 

112.36 


119.1016 amount at the end of 3 years. 

subtract 100 first principal. 

$19.10 interest for 3 years. Ans. 

Note. Instead of multiplying by .06, and adding the principal to the 
product for the amount, it is plain that we can obtain each amount by 
multiplying by 1.06. 

2. What will $255.16§ amount to in 5£ years, at 6 per 
cent., compound interest ; the interest being added to the 
principal in a new note at the end of each year ? 

Ans. $346.59. 

3. A man lent $1,000 at 7 per cent., on condition of re¬ 
ceiving the interest every 90 days ; what will be the whole 
amount in 1 year 40 days, if all the interest proceeding 
from this sum be lent on the same terms ? Ans. $1,081.24. 

4. What will a note for $1,843.12£ amount to in 6 
years, at 6 per cent., the interest being added to the prin¬ 
cipal in a new note at the end of each year ? 

Ans. $2,614.51. 


In example 1, instead of multiplying by .06 and adding the principal 
to the product for the amount, what course can we take ? 










160 


COMPOUND INTEREST. 


5. What will the interest on a note for $1,200 amount 
to in 3 years, at 6 per cent., the note being renewed every 
90 days, and the interest included in it ? Ans. $238.33. 

The following table will enable us to perform examples 
in compound interest with great ease. 


TABLE, 

Showing the amount of $1 at 5, 6, and 7 per cent., com¬ 
pound interest, from one year to 40. 


Years. 

At 5 per cent. 

At 6 per cent. 

| At 7 per cent. 

1 

$1.050000. 

$1.060000. 

$1.070000. 

2 

1.102500. 

1.123600. 

1.144900. 

3 

1.157625. 

1.191016. 

1.225043. 

4 

1.215506. 

1.262477. 

1.31079Q. 

5 

1.276282. 

1.338226. 

1.402552. 

6 

1.340096. 

1.418519. 

1.500730. 

7 

1.407100. 

1.503630. 

1.605781. 

8 

1.477455. 

1.593848. 

1.718186. 

9 

1.551328. 

1.689479. 

1.838459. 

10 

1.628895. 

1.790848. 

1.967151. 

11 

1.710339. 

1.898299. 

2.104852. 

12 

1.795856. 

2.012196. 

2.252192. 

13 

1.885649. 

2.132928. 

2.409845. 

14 

1.979932. 

2.260904. 

2.578534. 

15 

2.078928. 

2.396558. 

2.759032. 

16 

2.182875. 

2.540352. 

2.952164. 

17 

2.292018. 

2.692773. 

3.158815. 

18 

2.406619. 

2.854339. 

3.379932. 

19 

2.526950. 

3.025600. 

3.616528. 

20 

2.653298. 

3.207135. 

3.869684. 

21 

2.785963. 

3.399564. 

4.140562. 

22 

2.925261. 

3.603537. 

4.430402. 

23 

3.071524. 

3.819750. 

4.740530. 

24 

3.225100. 

4.048935. 

5.072367. 

25 

3.386355. 

4.291871. 

5.427433. 

26 

3.555673. 

4.549383. 

5.807353. 

27 

3.733456. 

4.822346. 

6.213868. 

28 

3.920129. 

5.111687. 

6.648838. 

29 

4.116136. 

5.418388. 

7.114257. 

30 

4.321942. 

5.743491. 

7.612255. 


What does the table show ? 








DISCOUNT. 


161 


Years. 

At 5 per cent. 

At 6 per cent. | 

| At 7 per cent. 

31 

4.538039. 

6.088101. 

8.145113. 

32 

4.764941. 

6.453387. 

8.715271. 

33 

5.003189. 

6.840590. 

9.325340. 

34 

5.253348. 

7.251025. 

9.978114. 

35 

5.516015. 

7.686087. 

10.676581. 

36 

5.791816. 

8.147252. 

11.423942. 

37 

6.081407. 

8.636087. 

12.223618. 

38 

6.385477. 

9.154252. 

13,079271. 

39 

6.704751. 

9.703507. 

13.994820. 

40 

7.039989. 

10.285718. 

14.974458. 


A table like this can be calculated by finding the amount 
of $1, at 5, 6 and 7 per cent., compound interest, for 1, 2, 
3, &c. years ; that is, by multiplying $1 by 1.05, 1.06, 
1.07, and the product by the same multipliers, and so on. 
See note to example 1. 

6. What is the compound interest on $50 for 30 years, 

at 7 per cent. ? Ans. $330.61. 

Explanation. What will $1 amount to in 30 years, at 
7 per cent., by the table ? Then what will $50 amount to ? 

7. What will the compound interest on $216 come to in 

19 years, at 6 per cent. ? Ans. $437.53. 

8. What will $18 amount to in 29 years 3 months, at 5 

per cent., compound interest ? Ans. $75.02. 

Explanation. What does $1 amount to in 29 years ? 
What does this sum amount to in 3 months ? 

9. What will $11 amount to in 39 years, at 6 per cent., 

the interest being added to the principal in a new note at 
the end of every year ? Ans. $106.74. 

10. What will the compound interest on $20.25 come to 
in 12 yrs. 2 mo. and 14 d., at 6 per cent. ? Ans. $21. 


DISCOUNT. 

Lesson 134 . 

1. A dealer in carriages has a chaise marked $195, but 
offers to make a discount or deduction of $23 ; what sum 
will purchase the chaise ? Ans. $172. 


How can a table like this be calculated ? 

14 # 







162 


DISCOUNT. 


2. A bookseller demanded $17.50 for Irving’s works 

handsomely bound, but concludes to discount 12 per cent., 
for immediate payment ; what is his cash price for the 
works ? Ans. $15.40. 

3. If you owe a man $500, to be paid in 2 years, with¬ 

out interest , what must you pay now to cancel the debt, if 
money is worth 7 per cent. ? Ans. $438.60. 

Explanation. It is plain that you must pay a sum, 
which at 7 per cent, interest will amount to $500 in 2 
years. The sum to be paid now may be considered as the 
principal, and $500 as the amount. You have therefore 
the rate, time, and amount given to find the principal. See 
rule in Interest, middle of lesson 130. 

The principal which will amount to a debt, not on inter¬ 
est, when it is payable, is called the present worth of that 
debt. The difference between the debt and present worth, 
is the proper discount to be made for immediate payment. 

4. What is the present worth of a note for $385 to be 

paid in 3^- years without interest, money being worth 6 
per cent. ? Ans. $318.18. 

5. A country trader bought $1,850.37^- worth of goods 

of a merchant of Philadelphia, on 6 months’credit ; what 
discount should the merchant make for immediate pay¬ 
ment, if money is worth 6 per cent. ? Ans. $53.89. 

Explanation. Get the present worth first. 

6. What is the present worth of a note for $9,825 to be 
paid in 276 days, if money is worth 7 per cent. ? 

Ans. $9,331.09. 

7. If you give a man one note for $400, not bearing in¬ 

terest, to be paid in 1 year, and another for $2,537.55, not 
bearing interest, to be paid in 2 years and 8 months, what 
sum must you pay down to cancel both of these notes, 6 
per cent, being the legal rate ? Ans. $2,564.90. 

8. A merchant paid $875 down for goods, and sold them 
the same day for $983,334, on 9 months’ credit ; what did 
he gain, money being worth 10 per cent. ? Ans. $39.73. 

9. If you are offered $3,000 down for a house, or $3,300 
to be paid in 2 years, without interest, which will be the 
best bargain, money being worth 6 per cent. ? 

Ans. $3,000 down is the best bargain. 

Explain how example 3 in Discount should be performed. 

What is called the present worth of a debt? What is the proper dis¬ 
count to be made for immediate payment ? 



BANKING. 


163 


10. A hardware dealer sold me a stove for $28.50, dis¬ 
counting 5 per cent, from the ordinary price for ready 
money ; what was the ordinary price ? Ans. $30. 

Explanation. See Percentage, example 10, lesson 121. 


BANKING. 

Lesson 135. 

A bank is an institution which trades in money. It is 
usually owned in shares by persons called stockholders, who 
choose a President and Directors to manage its concerns. 
The principal object of a bank is to make and lend notes, 
called hank bills, as money. 

When money is borrowed from a bank, the usual man¬ 
ner of proceeding is as follows. If A wants $1,000 for 
a certain time, say 90 days, and his friend B is willing to 
become his Surety, he writes a note promising to pay B 
or order $1,000 in 90 days ; B now indorses the note, 
that is, writes his name on the back, thereby making him¬ 
self a security for the payment. A then proceeds to the 
bank, and the officers, if they choose, discount the note, 
that is, they take it, cast the interest on $1,000 for 3 days 
more than the time, or 90 days, and deducting it from 
$1,000, hand A the balance in bills. The 3 days are 
called days of grace, and the bank does not require the 
$1,000 to be paid until the end of 93 days. Interest paid 
in this way is improperly called discount. It is plain, then, 
that banks take interest for larger sums than they lend, 
and by calling 30 days 1 month, 60 days 2 months, 90 days 
3 months, or £ of a year, 8lc., obtain interest for a longer 
time than the borrower has the money. 

1. Interest being 6 per cent., what sum do I get on a 
note for $683, payable in 60 days, which a bank discounts 
for me ? Ans. $675.83. 

Explanation. Remember the 3 days of grace. 

What is a bank ? How is it usually owned ? What is the principal 
object of a bank ? 

When money is borrowed from a bank what is the usual manner of 
proceeding ? What are the 3 days called, and when does the bank re¬ 
quire the note to be paid ? What in banking is called discount P What 
is plain ? 




164 


EQUATION OF PAYMENTS. 


2. What is the amount of the bank discount on the fol¬ 
lowing note, interest being reckoned at 7 per cent. ? 

$3^327,40. New York, April 20th, 1837. 

Ninety days from date I promise to pay James Carver 
or order, at the Mechanics’ Bank, three thousand three 
hundred and twenty-seven dollars and for value re¬ 
ceived George Stilman. 

Ans. $60.17. 

3. A merchant bought $10,000 worth of cotton at $50 
a bale, and sold it immediately at $60 a bale, obtaining a 
note for the amount of the sale, payable in 90 days, with¬ 
out interest ; what will be the amount of his gain, if he 

v gets the note discounted at a bank, 6 per cent, being the 
rate of interest ? Ans. $1,814 

4. How much did a man receive on a note for $982, 

payable in 70 days, which he had discounted at a bank, 
where the rate was 7 per cent.? Ans. $968.06. 

5. What is the bank discount on a note for $2,500, pay¬ 
able in 4 months, the rate of interest being 6 per cent. ? 

Ans. $51.25. 

6. What is the bank discount on a note for $700, pay¬ 

able in 90 days, where the legal rate of interest is 7 per 
cent.? Ans. $12.66. 

7. How much do I receive on a note for $2,000, pay¬ 

able in 30 days, which I get discounted at a bank, the rate 
of interest being 6 per cent.? Ans. $1,989. 

8. I have a note due me for $100, payable in 60 days, 

which the officers of a bank are willing to discount if I in¬ 
dorse it; what will the discount amount to, 6 per cent, being 
the rate of interest ? Ans. $1.05. 


EQUATION OF PAYMENTS. 

Lesson 136. 

1. A merchant owes a trading company the following 
notes. One for $8,480, due in 1 year, without interest, 
one for $1,526, due in 18 months, without interest, and one 
for $1,326, due in 21 months without interest; in what time 





EQUATION OF PAYMENTS. 165 

can these debts be paid at once, and neither party sustain 
loss, reckoning any per cent., say 6 ? 

Ans. 1 yr. 1 mo. 24 d., about. 

Explanation. We first find the present worth of each 
of these notes, and then find in what time the sum of these 
present worths amounts to the sum of the debts. This 
time is evidently the answer. 

Therefore, to find the time when several debts, due at 
different times, can be paid at once without loss to either 
debtor or creditor, 

Find the present worth of each of the debts , and then find 
in what time the sum of these present worths icill amount to 
the sum of the debts. This time will be the answer. 

2. A merchant sold a country trader a quantity of goods, 
which were to be paid for as follows ; $2,500 were to be 
paid in 4 months, $350 in 6 months, and $1,000 in 8 
months ; in what time can all these sums be paid at once, 
without loss to either party ? Ans. in 5 mo. 6 d., about. 

3. I bought goods to the amount of $10,067, agreeing 
to pay $2,030 at the end of 3 months, $2,575 at the end of 
6 months, $2,600 at the end of 8 months, and $2,862 at the 
end of 12 months ; it was afterwards thought best that the 
whole amount should be paid at once ; how long after the 
purchase should the payment be made ? 

Ans. 7 mo. 17 d., about. 

Other Questions concerning the Payment of Debts. 

4. If you owe a man a note for $80, on demand, with 

interest, and three other notes not drawing interest, one 
for $16, due in 2 months, one for $25, due in 3 months, 
and one for $40, due in 6 months, what sum must you pay 
at the end of 4 months to discharge the whole, 7 per cent, 
being the legal interest ? Ans. $162.74. 

Explanation. A note not on interest is considered as 
drawing interest as soon as it becomes due; this being un¬ 
derstood, we find the amount of $80 for 4 months, of $16 for 
2 months, of $25 for 1 month, and the present worth of $40 


Explain how example 1 in Equation of Payments is performed. 

How do we find the time when several debts, due at different times, 
can be paid at once without loss to either debtor or creditor P 
When is a note, not on interest, considered as drawing interest? 
Explain how example 4 in Equation of Payments is performed. 



166 


EQUATION OF PAYMENTS. 


paid 2 months before due. These four sums added to¬ 
gether will evidently give the answer. 

5. I owe a man §600, due in 1 year without interest, 

$800 due in 2 years without interest, and he has my note 
for $2,000, on demand, with interest ; what sum should I 
pay to cancel these debts at the end of 4 months, the rate 
being 6 per cent.? Ans. $3,344.19. 

6. If I owe you $1,200, to be paid at the end of 6 months, 
but to accommodate you pay $400 down, how long must I 
keep the remainder so that neither of us shall lose interest ? 

Ans. 9 mo. 1 d., about. 

Explanation. What is the amount of $400 on interest 
for 6 months ? This amount being subtracted from $1,200, 
how long will it take the remainder to amount to $800 ? 

7. A farmer had $200 due him in 1 year, but the debtor 

chose to pay $100 down, and $30 more at the end of 8 
months ; what time should he be allowed to keep the re¬ 
maining $70 ? Ans. 1 yr. 11 mo. 25 d., about. 

Another common hut incorrect method. 

To find the time when several debts, due at different 
times, can be paid at once, without loss to either debtor or 
creditor, merchants usually 

Multiply each debt hy the time to elapse before it is due , 
and divide the sum of the products by the sum of the debts. 

Though this method is inaccurate it is easy, and experi¬ 
ment shows it may be employed without much error, to 
find the mean time for the payment of a number of debts, 
the first and last of which are due within a year of each 
other. The reasoning employed in many arithmetics to 
show its accuracy is absurd. 

Perform the following examples by this rule. 

8. Example 2, lesson 136. Ans. 5 mo. 7 d., nearly. 

9. Example 3, lesson 136. Ans. 7 mo. 19 d., nearly. 

10. A. B. sells the following articles, on 6 months’ credit 
to C. D., who on May 13th, 1831, settles with A. B. by 
giving his note, not on interest, for the whole sum due ; 

What course do merchants usually take to find the time when several 
debts, due at different times, can be paid at once without loss to either 
debtor or creditor ? 

When may this method be employed ? What is said of the reasoning 
concerning it ? ^ 



PROMISCUOUS QUESTIONS. 


167 


when must the note be made payable so that no interest 
will be gained or lost by either party ? 

Dr. C. D. 

1831. 

Jan. 4. To 17 bis. Flour, at $8 a bl. $136 

Feb. 28. “ 108 gals. Molasses, at 25 cents a gal. 27. 
April 7. “ 200 lbs. Malaga Raisins, at 6£ cents a lb. 12.50 
“ 19. “ 6 bis. Mackerel, at $12 a bl. 72. 

Ans. in 94 d., or August 15th, 1831. 


PROMISCUOUS QUESTIONS 

IN 

PERCENTAGE, COMMISSION, STOCKS, AND THE SIMILAR 
RULES. 

Lesson 137. 

To be performed in the mind. 

1. What is the interest on $200 for 2 y§ars, at 5 per 
cent. ? At 6 per cent. ? At 8 per cent. ? At 12 per 
cent. ? 

2. I borrowed $100 for one year, at 6 per cent., but 

kept the sum 1 year and 3 months ; what interest was 
then due ? What interest would have been due had I kept 
the money 1 year and 4 months ? 1 year and 6 months ? 
1 year and 10 months ? . 

3. A merchant bought some St. Domingo hides for 
$450, and immediately sold them so as to gain 3 per cent. ; 
how much did he gain ? 

4. What is the bank discount on a note for $80 payable 
in 60 days, the rate of interest being 6 per cent. ? 

Explanation. What is the interest of $80 for 60 days, 
and for the 3 days of grace, being of 60 days ? 

5. A bankrupt who pays only 25 per cent, of his debts 
owes me $41, what sum shall I get ? 

Explanation. See Decimal Fractions, lesson 82. 

6. Calling 30 days a month, what will the interest of 
$100 amount to in 15 days, at 6 per cent. ? In 20 days ? 
In 25 days ? 

7. What duty must I pay on 75 tons of English iron, at 
$30 a ton ? 



168 


PROMISCUOUS QUESTIONS. 


8. My agent bought me a farm for 62,000 ; if I pay 
him 3 per cent, commission for his services, what sum 
must he receive ? 

9. The shares in a certain manufacturing company, are 
worth 6500 a piece at par ; what must I give for 2 shares 
at 4 per cent, discount ? 

10. What is the interest of 690 for 3 months, at 5 per 
cent. ? 

11. What is 8 per cent, of 64,000 ? 7 per cent. ? 

12. What is the compound interest on 650 for 2 years, 
at 6 per cent. ? 


Lesson 138 . 

For the Slate. 

1. What is the interest on a note for 6100 dated July 
18th, 1830, and paid November 3d, 1831, at 5f- per cent. ? 

Ans. 67.45. 

2. A man had 6600 due him in 3 months, without inter¬ 

est, and 6375 due in 1 year 133 days, without interest ; 
what sum must be paid in 2 months to cancel these debts, 
6 per cent, being allowed ? Ans. 6946.82. 

3. If 8 per cent, be deducted from the amount of a bill, 

and the creditor be paid 6276, what is the amount of the 
bill ? Ans. 6300 

4. Received by the brig Nestor, 30 pipes of Port wine, 

found by the gauges to contain 3,900 gallons ; what is the 
amount of duty to be paid, at 15 cents a gallon, allowing 
for leakage ? Ans. 6573.30. 

5. A man paid 612.64 for the use of 6200 during 1 year 
3 months and 5 days ; what was the rate given ? 

Ans. 5 per cent. 

6. An agent received 3 per cent, for selling 15 lbs. 10 

oz. of silver in ingots, at 613 a pound ; what did his com¬ 
mission amount to ? Ans. 66.17. 

7. 61,200 amounted to 61,216.80 in a certain time, at 7 

per cent. ; what was the time ? Ans. 73 days. 

8. 8 shares in the Boston and Worcester railroad sold 
for 626.66§ less than the par value, which is 6100 a share ; 
at what per cent, below par did they sell ? 

Ans. 3£ per cent. 

9. What principal will amount to 681.20 in 2 years 8 

months, at 6 per cent. ? Ans. 670. 

10. What sum shall I get at a bank for a note for 6800 

payable in 90 days, at 6 per cent. ? Ans. 6787.60. 


PROMISCUOUS QUESTIONS. 


169 


Lesson 139. 


1. A trader bought a quantity of goods for $2,000, ana 
marked the price 25 per cent, more than the cost ; what 
per cent, must he deduct from the price marked, in order 
to gain 10 per cent. ? _ Ans. 12 per cent. 

$350. Steubenville, May 4th, 1827. 


For value received, I promise to pay John Darnell, or 
order, three hundred and fifty dollars, on demand, with 
interest. William Tolman. 

On the back of this note were the following endorse¬ 
ments ; 

January 11th, 1828, received $50. 

June 13th, 1828, received $25. 

What was due November. 12th, 1828, 6 per cent, being 
allowed ? Ans. $304.89. 

3. If you owe a man $400, due in 9 months, without in¬ 

terest, and pay him $150 down, how long should he wait 
for the remainder ? Ans. 14 mo. 16 d., about. 

4. A merchant imported manufactured articles from 
Liverpool, which cost at that place $4,000 ; what will the 
duties amount to, at 20 per cent, ad valorem ? 

Ans. $800. 

5. What will $700 amount to in 35 years, at 7 per cent., 

compound interest ? Ans. $7,473.61. 

6. I owe the following notes, not bearing interest ; one 
for $250.25, due in 6 months, one for $11.30, due in 9 
months, one for $65, due in 15 months, and one for $110, 
due in 18 months ; in what time must I pay all these 
notes so that neither I nor my creditor shall lose ? 

Ans. in 10 mo. 9 d., about. 

7. If I owe a debt of $580, due in 1 year 3 months, 

without interest, what sum should I pay now to cancel the 
debt, money being worth 6 per cent. ? Ans. $539,53. 

8. What sum was due on the following account at inter¬ 
est, at 6 per cent., settled January 1st, 1835. 

George Draper’s cash account with S. Pierce. 

Dr. 


1834. 

May 15. To Cash,.... $400 
Nov. 2. “ Cash,.... 1,000 


1834. 
April 3. 
Aug. 31. 


Cr. 

By Cash. .. .$800 
“ Cash.900 


Ans. $329.05 due George Draper. 


15 






170 


RULE OF THREE. 


RULE OF THREE. 

So called because questions in this rule usually have 
three numbers given to find another, or the answer. This 
rule embraces many questions of so different a nature that 
no general rule can be given for calculating them. 

Lesson 140. 

To be performed in the mind . 

1. John gave 10 cents for 1 pound of sugar ; how many 
cents must he give for 3 pounds ? For 5 pounds ? For 8 
pounds ? For 9 pounds ? For 12 pounds ? For 15 pounds ? 

2. A teamster hauled 24 cwt. of pork in 4 equal loads ; 
how much did he haul in 1 load ? If he had hauled 28 
cwt. in 4 equal loads, how much would there have been in 
1 load ? If he had hauled 30 cwt. in 5 equal loads, how 
much would there have been in each load ? 

3. If you get 6 apples for 3 cents, how many can you 
obtain for 18 cents ? 

Explanation. How many apples can you obtain for 1 
cent ? How many, then, can you get for 18 cents ? 

4. If you pay 9 cents for 3 oranges, how many cents 
must you pay for 8 oranges ? 

Explanation. How many cents do you pay for 1 or¬ 
ange ? How many cents, then, must you pay for 8 or¬ 
anges ? 

5. A farmer bought 2 gallons of molasses for 60 cents, 
what would 5 gallons have cost ? 

6. If you buy 4 bushels of potatoes for 2 dollars, how 
many bushels can you get for $30 ? 

7. A woman bought 3 yards of cotton cloth for 60 cents ; 
how many yards could she have bought for $2, or 200 
cents ? 

8. Suppose a man travels 40 miles in 5 days, how far 
can he travel, at the same rate, in a week, or 7 days ? 

9. If 16 tons of hay keep 8 cows over winter, how many 
tons will keep 5 cows ? 

10. A man gives $45 for 3 loads of hay ; how much 
must he give for 7 such loads ? 


Why is the Rule of Three so called ? What does this rule embrace ? 




RULE OF THREE. 


171 


11. If 4 barrels of pork weigh 8 hundred pounds, how 
much will 9 barrels weigh ? 

12. A man bought 8 yards of cloth for $4 ; how many 
yards can he buy, at the same rate, for $25 ? 

Lesson 141 . 

1. A certain load of hay will last 6 oxen 5 days ; how 
long will it last 1 ox ? 

2. If 1 pipe will empty a cistern in 12 hours, how long 
will it take 3 such pipes to empty it ? 

3. If 1 pipe will empty a cistern in 12 hours, how many 
of such pipes will empty it in 2 hours ? 

Explanation. How many will empty it in 1 hour ? How 
many, then, will empty it in 2 hours ? 

4. If 1 man can do a piece of work in 16 days, how 
many men will it take to do it in 4 days ? 

5. 8 men have provisions enough to last them 9 days ; 
how long will the provisions last 12 men ? 

Explanation. How long will the provisions last 1 man ? 
How long, then, will they last 12 men ? 

6. 6 men have provisions enough to last them 8 days ; 
how many men will these provisions last 12 days ? 

Explanation. How many men will these provisions last 
1 day ? How many, then, will they last 12 days ? 

7. A teamster can carry a certain number of barrels of 
beef in 4 loads if he takes 6 at a load ; how many loads 
must he make if he takes 8 barrels at a load ? 

8. If a certain quantity of oats last 4 horses 3 days, how 
long will it last 3 horses ? 

9. 2 oxen can haul away a pile of lumber in 20 loads ; 
what number can haul it away in 5 loads ? 

10. A garrison of 200 men have provisions enough for 
12 months ; if the number be increased to 600, how long 
will the provisions last ? 

11. If 6 men can do a piece of work in 2 days, how 
many men will be necessary to complete it in 4 days ? 

12. If 6 equal sized pipes will empty a cistern in 2 
hours, how many pipes of the same magnitude will empty 
it in 3 hours ? 


172 


RULE OF THREE. 


Lesson 142. 

1. If 4 apples cost 3 cents, what will 12 apples cost ? 

2. What will 5 pounds of raisins cost, if 2 pounds cost 
15 cents ? 

3. If 3 oranges cost 6 cents, how many will 24 cents buy ? 

4. A man gave $4 for £ of a yard of broadcloth ; how 
much would £ of a yard have cost ? 

Explanation. How much would 1 yard have cost ? Then 
how much would £ of a yard have cost ? 

5. James gave 6 cents for £ of a pound of raisins ; how 
many pounds could he have bought for 36 cents ? 

6. What will 5 gallons of molasses cost, if 1 quart, or £ 
of a gallon is worth 10 cents ? 

7. 12 men dug a ditch in £ of a day ; in what time 
could 2 men have dug it ? 

8. 8 men can do a piece of work in £ of a day ; how 
many men will it take to do it in § of a day ? 

9. If f of a vessel cost $4,000, how much will £ of it 
cost ? 

10. A man bought 4 yards of shirting for 48 cents ; how 
much would £ of a yard have cost at this rate ? 

It. If 3 quarts of beans cost 37£ cents, what will 2 
quarts cost ? 

12. A boy gave 13 cents for 2 picture books; how many 
books of the kind could he have purchased for 65 cents ? 

Lesson 143. - 

For the Slate . 

1. If 18 tons of hay cost me $324, what will 23 tons cost, 

at the same rate ? Ans. $414. 

2. 4 horses being found to consume thirty bushels of oats 

in 6 weeks, how many horses will consume the same quan¬ 
tity in 4 weeks ? Ans. 6. 

3. If I give $2 for .4 of a cord of wood, how much must 

I give for £ of a cord ? Ans. $2.50. 

4. What will 19 barrels of flour cost, if 4 barrels cost 

$26 ? Ans. $123.50. 

5. If 19 barrels of flour cost $123.50, what will 4 bar¬ 
rels cost ? Ans. $26. 

6. A man gave $125 for 5 barrels of pork ; how many 

Darrels could he have got for $275. Ans. 11. 


RULE OF THREE. 


173 

7. A stone wall was built in 12 days by a number of 

workmen who labored only 6 hours in a day ; how long 
would it have taken them if they had labored 8 hours in a 
day ? Ans. 9 days. 

8. A merchant, who owned f of a ship, sold f of his 

part for $7,500 ; what was the whole ship worth at this 
rate ? Ans. $14,062.50. 

Explanation . What part of the ship did he sell ? 

9. A. and B. hired a pasture for $35.20, in which A. 
pastured 15 oxen and B. 7 ; what should each one pay ? 

Ans. A. should pay $24, and B. $11.20. 

Explanation. If the pasturing of 22 oxen cost $35.20, 
what will the pasturing of 15 cost, and of 7 cost ? 

10. If a man pays $28.12£ for 18 bu. 3 pks. of wheat, 
what can he get 10 bu. for, at the same rate ? Ans. $15. 

Lesson 144. 

1. If I give $15 for 7 yards of cloth, what must I give 
for 1,235 yards, at the same rate ? 

OPERATION. OPERATION BY AN EASIER METHOD. 

7)15 $4^ price of 1 yard. 

- 1 2 3 5 yards. 

2.1 4 2 8 6 price of 1 yard. 1 5 

1 2 3 5 yards. - 

- 6 17 5 

1071430 1235 

642858 - 

428572 7)18525 

214286 - 

- $2 6 4 6.4 3 Ans. 

$2 6 4 6.4 3 2 1 0 price of 1,235 yards. Ans. 

Explanation. To obtain the answer we get the price of 
lyard, and multiply it by 1,235, the number of yards; now 
if we get the price of 1 yard in dollars, cents, mills, &c., 
by dividing $15 by 7, and then multiply this price by 1,235, 
the operation, as we see, is tedious; but if we get the price 
of 1 yard in the form of a fraction, by writing 7 under $15, 
according to Common Fractions, lesson 60, and then mul¬ 
tiply $ J T - and 1,235 together, the operation is much easier. 

TVe should ahoays get a quotient in the form of a fraction 
whenever an operation by that course becomes easier. The 

Explain how example 1, lesson 144, can be performed. 

When should we get a quotient in the form of a fraction ? 

15* 









174 


RULE OF THREE. 


other method is the best when the number of figures in 
the quotient will be less than the number of figures in 
both divisor and dividend. 

2. The discount on a note for $53, due in one year, 
without interest, being $3, what will be the discount on a 
note for $18,232, due in the same time, without interest ? 

Ans. $1,032. 

3. If 5 yards of cloth cost $6f, what will 12§ yards cost ? 

Ans. $16.15. 

4. A grocer gave $20 for 6 barrels of apples; how many 

barrels could he have obtained for $70 ? Ans. 21. 

5. 16 men can dig a ditch in 9 days ; how long will it 

take 6 men to dig it ? Ans. 24 days. 

6. If 12 soldiers can subsist on 17 pounds of provisions 

1 day, what quantity will suffice for 2,300 men for the 
same time ? Ans. 3,258£ lbs. 

7. A blacksmith gave $5.60 for 1 cwt. of Swedes iron ; 

how much must he pay for 15 cwt. 2 qrs. 8£ lbs., at the 
same rate ? Ans. $87.21. 

8. A farmer gave $838.56 for 83 A. 3 qrs. 17 sq. rods 

of land ; for what sum can he sell 5 A. 2 qrs. 38 sq. rods, 
so as neither to gain nor lose ? Ans. $57.37^. 

9. If a trader pays $!8 for a hogshead of molasses, 

containing 80 gals., what should he pay for 2 hhds. 1 tier. 
23 gals.? Ans. $42.97£. 

10. 16,000 cubic feet of water were found to flow over 

a mill-dam in 1 h. 23 min. 10 sec.; how much will flow 
over the dam in 3 days ? Ans. 831,102^§§ cubic ft. 

Lesson 145. 

Note. Many questions in arithmetic, contain numbers that are of no 
use in the calculation. In what precedes, we have been careful to 
write such numbers in words, but shall hereafter express them in 
figures; and the ingenuity of the learner must be exercised in discover¬ 
ing what numbers must be employed to obtain the answer. 

1. 3 men laid 28 rods of stone wall in 5 days ; how long 
then should it take 10 men to lay the same amount ? 

Ans. 1^ day. 

2. A merchant who owned § of a block of stores, sold % 

of his share to 5 men, for $12,000; what was the whole 
block worth, at this rate ? Ans. $72,000. 

3. If 4£ yards of broadcloth cost $17, what will 131- 

yards cost ? Ans. $55.50. 


When will the other method be the best ? 



RULE OF THREE. 


175 


4. A merchant bought 3 tierces of rice, each of which 
contained 6 cwt. 3 qrs. 27 lbs., for $135 ; how many 
pounds could he have purchased for $87.33^ ? 

Ans. 1,519.6 lbs. 

5. 5 tons of hay will last 80 sheep 120 days ; how long 

will it last 150 sheep ? Ans. 64 days. 

6. A trader hired a man to team 6£ tons of iron 28 miles, 
for $10 ; how many modern tons weighing 2,000 lbs. 
apiece, can he get teamed 16 miles for the same sum ? 

Ans. 12£ modern tons. 

7. If 20 bushels of wheat can be obtained for 100 bush¬ 

els of potatoes, how much wheat can be obtained for 575 
bushels of potatoes ? Ans. 115 bushels. 

8. If you lend a person $400 for 9 months, with the 

promise of having a like favor, and some time after borrow 
$600 of him, how long should you keep that sum in order 
to receive full compensation ? Ans. 6 months. 

9. A trader bought a quantity of cloth estimated at 88 

yards, for $169 ; on measuring it he found there were 97 
yards ; if he sells it so as to make $25, for what sum must 
he sell 5 yards ? Ans. $10. 

10. A grocer sells coffee at $.33£ a pound, and gains 
25 per cent. ; how much did 11 pounds cost him ? 

Ans. $2.93£. 


Lesson 146. 

1. A man gave $5.30 for 6.25 cwt. of hay ; how much 
would 1 ton have cost at the same rate ? Ans. $16.96. 

2. A trader having £ of a cwt. of coffee, sold § of it for 
$19.37£ ; at what price a pound did he sell it ? 

Ans. $.29656, nearly. 

3. If $13.50 will buy .75 of a boat, what sum will buy 

.33£ of her ? Ans. $6. 

4. 112£ yards of cloth, 1^ yard wide, will make 25 suits 

of clothes; how many yards of cloth f of a yard wide, will 
make the same number of suits ? Ans. 225. 

5. If 914 yards of cloth, £ of a yard wide, are sufficient 

to make 10 suits of clothes, what quantity of cloth, 1£ 
yard wide, will be required to make the same number of 
suits ? Ans. 45£ yds. 

6. The moon moves 13° 10' 35" towards the east in 1 
day ; how long is it in performing a revolution round the 
earth, that is, how long is it in passing over 360° ? 

Ans. 27 d. 7 h. 43 min. 6 sec. 


176 


RULE OF THREE. 


7. If .75 of a pound of butter costs $.1875, what quan¬ 
tity will $37 buy ? Ans. 148 lbs. 

8. A trader bought f of a pipe of wine for $100 ; what 
sum should he give for 3 tierces of the same quality ? 

Ans. $133.33£. 

9. If 3 C. 5 ft. of oak wood are of the same value as 1 

T. 5 cwt. 3 qrs. of hay, what quantity of wood should 1 
modern ton of hay purchase ? Ans. 2 C. 4^ ft. 

10. If a bar of iron 7 feet long, 4 inches wide, and 1fl¬ 

inch thick, weighs 140 pounds, what will a bar 3 ft. 2 in. 
long, and 2 in. square weigh Ans. 39 lbs. 7^ oz. 

Explanation. How many cubic inches weigh 140 pounds. 


Lesson 147. 

1. Tf 4 men can lay a w T all 75 feet long, 4 feet thick, and 
10 feet high, in 8 days, how long will it take them to lay a 
wall 97 feet long, 3^ feet thick, and 9 feet high ? 

Ans. 8.148 d 

2. If 125 bushels of wheat grow on 4 A. 2 qrs. 4 sq, 
rods, how much land will be necessary to produce 650 
bushels, supposing the crop to be equally good ? 

Ans. 23 A. 2 qrs. 4.8 sq. rods, 

3. Bought 10 gals. 3 qts. 1 pt. of wine for $16.16§ ; 

what sum would 27 gals. 2 qts. have cost, at the same 
rate ? Ans. $40.88. 

4. A merchant bought a hogshead of sugar, containing 

5 cwt. 3 qrs. 7£ lbs., for $50 ; what should he give for 2 
other hogsheads, at the same rate, one of which contains 

6 cwt. 18 lbs., and the other 4 cwt. 3 qrs.? Ans. $93.82. 

5. The fore wheels of a wagon being 11 ft. in circum¬ 

ference, turn round 960 times in' 2 miles ; how many times 
do the hind wheels, which are 14 ft. in circumference, turn 
round in the same distance ? Ans. 754f times. 

6. A wine merchant bought 64 pipes of wine for $4,000; 
he paid $58 for freight, $29 duties, $20 for storage, and 
for truckage and other expenses, $35 ; if he sells it so as 
to make $500, for what sum can I get 13 pipes ? 

Ans. $942.91. 

7. If a man buys 257 chal. 15 bu. 3 pks. of Sidney coal 

for $2,000, and sells it so as to make 25 per cent., for 
what sum should he sell 13^-chal. ? Ans. $131.10. 

8. A trader bought 10 puncheons of rum for $800 ; 2 


RULE OF THREE. 


177 


puncheons were stove, through accident, and he lost 28 
gals, besides, from leakage ; if he sells the remainder so 
as to lose 9 per cent, of the whole cost, for what sum can 
you buy 3 puncheons ? Ans. $284.87. 

9. If 3,000 lbs. of provisions last 24 men 8 weeks and 4 
days, how long will the same quantity last 96 men ? 

Ans. 2 w. Id. 

10. If 3.25 lbs. of tobacco cost $.975, what quantity can 

you purchase for $5.46 ? Ans. 18.2 lbs. 

Lesson 148. 

1. A. can reap a field in 6 days, and B. in 10 days; how 
many days will it take both of them to reap it ? Ans. 3£ d. 

Explanation. What part of it can A. reap in 1 day ? 
what part of it can B. reap in 1 day ? What part of it 
then can A. and B. both reap in 1 day ? How long will it 
take them both to reap it then ? 

2. A cistern has 4 pipes ; the first will empty it in 12 

minutes, the second in 15 minutes, the third in 30 minutes, 
and the fourth in 45 minutes ; in what time will they all 
empty it ? Ans. 4 min. 51$^- sec. 

3. How long will you be in overtaking a horseman, who 

has been gone 5 hours, and who rides 8 miles an hour, if 
you ride 10 miles an hour ? Ans. 20 h. 

4. If a man who owes $3,000, drawing 6 per cent, in¬ 
terest, receives $4.50 a day, and spends $1.45 a day, how 
long before he will be worth $1,000 ? 

Ans. 4 yrs. 104 d., about. 

5. The hour and minute hands of a watch are together 
at 12 o’clock ; when are they together again ? 

Ans. in 1 h. 5 min. 27 -fa sec 

Explanation. Observe that the minute hand travels 60 
minutes in an hour, and the hour hand 5 minutes in an hour. 

6. A. can mow a piece of ground in 11 days ; with the 

assistance of B. he can do it in 7 days ; in what time can 
B. do it alone ? Ans. in 19£ d. 

7. 2 men, A. and B., being 50 miles apart, commenced 
travelling from each other ; A. proceeds at the rate of 2.75 
miles an hour, and B. at the rate of 3.5 miles an hour ; 
how long before they will be 110 miles apart ? 

Ans. in 9 h. 36 min. 

8. A. lives in New York, and B. in Philadelphia ; A. 
can travel £ of the distance between the 2 cities in 4 hours, 


178 


RULE OF THREE COMPOUND. 


and B. can travel £ of the distance in 5 hours ; if they 
both start to travel towards each other, at the same mo¬ 
ment, how long before they will meet ? Ans. in 12 h. 

9. If 1 T. 2 cwt. 1 qr. of hay will last 3 horses 4 days, 

what quantity will be necessary to last 7 horses the same 
length of time ? Ans. 2 T. 11 cwt. 3§ qrs. 

10. A man owns 4 farms ; the first contains 200 A., and 
the second 250 A. ; the other 2 farms bear the same propor¬ 
tion to each other as to size, but the fourth contains only 
31 A. ; how much land is there in the third farm ? 

Ans. 24 A. 3 qrs. 8 sq. rods. 


RULE OF THREE COMPOUND, 

CALLED ALSO 

DOUBLE RULE OF THREE, AND COMPOUND PROPORTION 

Lesson 149 . 

Questions in the Rule of Three Compound, contain two 
questions in the Rule of Three, and the learner must ex¬ 
ercise his judgment in decomposing them, and in obtain¬ 
ing several partial answers from which to deduce the true 
answer. 

1. If it takes 2 years for the interest of $150 to amount 
to $18, how long will it take for the interest of $675 to 
amount to $162 ? Ans. 4 yrs. 

Explanation. How long will it take for the interest of 
$1 to amount to $18 ? How long will it take for the in¬ 
terest of $1 to amount to $1 ? How long will it take for 
the interest of $675 to amount to $1 ? How long then will 
it take for the interest of $675 to amount to $162 ? 

Note. There are many ways to decompose a question, and obtain 
the answer by methods similar to the preceding. Recollect that when¬ 
ever we have to divide one number by another, we can write the di¬ 
visor under the dividend, and form a fraction, which can be used as the 
qurrtient. When this course renders an operation easier, it should be 
adopted. 


What is said of questions in the Rule of Three Compound ? 

What ways are there to decompose a question in the Rule of Three 
Compound, and obtain the answer ? What is to be recollected ? When 
should this course be adopted t 




RULE OF THREE COMPOUND. 


179 


2. The interest on $200 for 4 months being $4, what 

will be the interest on $590 for 1 year and 3 months, that 
is, for 15 months ? Ans. $44.25. 

3. If 76 bushels of oats last 19 horses 16 days, how long 

will T2 bushels last 3 horses ? Ans. 16 days. 

4. A trader paid $3 for the transportation of 14 cwt. 
2 qrs. 17 lbs. 42 miles ; how much ought he to pay for the 
transportation of 6 T. 26 miles, at the same rate ? 

Ans. $15.21. 

5. 4 men built a wall 33 feet long, 14 feet high, and 5 
feet thick, in 13 days ; how long then should it take 7 men 
to build a wall 40 feet long, 2£ feet thick, and 7 feet high ? 

Ans. 2^ d. or 2f d., about. 

6. If 3 lbs. of rice cost 24 cents, and 12 lbs. of rice are 

worth as much as 5 lbs. of coffee, what are 78 lbs. of cof¬ 
fee worth ? Ans. $14.98. 


Lesson 150. 

1. By working 8 hours in a day, 7 men hoed 21 acres in 

6 days ; how long then will it take 4 men, working 11 
hours in a day, to hoe 164 acres ? Ans. 6 days. 

2. 12 men put f of a vessel’s cargo on board in £ of a 

day ; how long then will it take 5 men to load 3 such ves¬ 
sels ? Ans. 13£ d. 

3. If 15 bushels of wheat last 12 men 75 days, how long 

will 45 bushels last 30 men ? Ans. 90 d. 

4. A garrison of 400 men had a supply of provisions for 

32 weeks, each individual receiving 32 ounces a day ; at 
the end of 20 weeks, 50 of the men were killed in an as¬ 
sault ; what quantity of food can each of the remaining 
men receive a day, so that the provisions may last 30 
weeks longer ? Ans. 14§f oz. 

5. If f of a yard of cloth, f of a yard wide, cost $2f, 

what will 34 yards of cloth of the same quality cost, that 
is If yard wide ? Ans. $29.40. 

6. A man gave $15.50 for .875 of a load of hay ; what 
should he pay for .8 of another load of the same quality, 
estimated to contain .75 as much hay as the first ? 

Ans. $10.63 

7. If 3 men are hired 6 days for $21, how many men 
can be hired 15 days, at the same rate, for $157.50 ? 

Ans. 9 


180 


CHAIN RULE. 


CHAIN RULE, 

CALLED ALSO CONJOINED PROPORTION. 

Lesson 151. 

Questions in the Chain Rule are a continuous chain of 
questions in the Rule of Three. 

1. If 2 lbs. of butter are worth as much as 3 lb3. of 
cheese, and 8 lbs. of cheese are worth as much as 4 lbs. 
of tea, how many lbs. of butter can be obtained for 6 lbs. 
of tea ? 

Operation and Explanation. It is plain that 1 lb. of 
cheese is worth § of a lb. of butter, and that 1 lb. of tea is 
worth f of a lb. of cheese ; 1 lb. of tea, then, is worth f- of 
| of a lb. of butter, or -ff of a lb. of butter, and 6 lbs. of 
tea are worth 6 times of a lb. of butter, or ff of a lb. of 
butter, or 8 lbs. of butter. 

2. If 2 lbs. of butter are worth as much as 3 lbs. of 
cheese, and 8 lbs. of cheese are worth as much as 4 lbs. of 
tea, how many pounds of tea can be purchased for 12 lbs. 
of butter ? 

Operation and Explanation. It is plain that 1 lb. of but¬ 
ter is worth § of a lb. of cheese, and that 1 lb. of cheese is 
worth f of a lb. of tea ; 1 lb. of butter, then, is worth § of 
£ of a lb. of tea, or -ff of a lb. of tea, and 12 lbs. of butter 
are worth 12 times of a lb. of tea, or of a lb. of tea, 
or 9 lbs. of tea. 

From the preceding, we obtain the following rule. 

To find what quantity of the first thing mentioned is 
equal in value to a certain quantity of the last, 

Multiply together the jirst number in the question , and 
every alternate one , and divide this product by the product of 
the other numbers. 

To find what quantity of the last thing mentioned is 
equal in value to a certain quantity of the first, 

Multiply together the first number in the question , and 

What are questions in the Chain Rule ? 

State how example 1, lesson 151, is performed. 

State how example 2, lesson 151, is performed. 

What is the rule we obtain ? 



BARTER. 


181 

every alternate one , except the last , and divide by this product 
the product of the other numbers. 

3. 6 Massachusetts shillings are equal to 8 New York 

shillings, and 16 New York shillings are equal to 15 Penn¬ 
sylvania shillings, and 30 Pennsylvania shillings are equal 
to 20 Canada shillings ; now how many Massachusetts shil¬ 
lings are equal to 40 Canada shillings ? Ans. 48. 

4. If $31 in the United States are equal to 25 milrees in 

Portugal, and 10 milrees in Portugal are equal to 31 
florins in Amsterdam, how many florins in Amsterdam are 
equal to $16 in the United States ? Ans. 40. 

5. If 25 boys can do as much as 12 men, and 10 men 

can do as much as 15 women, how many women will be 
necessary to dq the work of 50 boys ? Ans. 36. 

Note. The rules in this lesson are not indispensable; the learner 
should now perform the examples without employing them. 


BARTER, 


OR THE EXCHANGE OF COMMODITIES. 

Lesson 152. 

To be performed in the mind. 

1. A farmer bought 1 tierce of molasses, at $.40 a. gal¬ 
lon, and agreed to pay in corn at $.80 a bushel ; what 
quantity will discharge the debt ? 

2. How much wood at $8 a cord can I get for 2 tons of 
hay at $18 a ton ? 

3. A farmer in Pennsylvania sold a quarter of veal 
weighing 18 lbs. at 9 cents a lb., and received in payment 
10 lbs. of rice at 5 cents a lb., a levy’s worth of raisins, and 
the rest in cash, how much cash did he get ? 

4. If a trader in Portland sells you a straw bonnet for 
15 shillings, and receives in payment 3 pairs of stockings at 
2 shillings a pair, 2 pairs of mittens at 15 cents a pair, and 
the balance in money, how much money do you pay him ? 

5. How much butter at 1 shilling a pound, must I give a 
trader in Troy, New York, for a quantity of shad worth 
$2.50. 


16 



182 


ASSESSMENT OF TAXES. 


For the Slate. 

6. What quantity of cod fish, at $3 a quintal, must be 
given for 19 cords 7 ft. of wood, at $6 a cord ? 

Ans. 39 quintals 84 lbs. 

7. How many bushels of potatoes, at $.4If a bushel, 

must be given in payment for 12 yards of broadcloth, at 
$4.31£ a yard ? Ans. 124.2 bu. 

8. What quantity of cheese, at $.12f a pound, must be 

given in exchange for 1 hogshead of St. Ubes salt, valued 
at $4.75 ? Ans. 38 lbs. 

9. A farmer exchanging some, eggs for a quantity of 

coffee, found that the grocer demanded 36 cents a pound 
for coffee that was worth only 30 cents, in cash ; what 
price ought he to ask for his eggs, which were worth 15 
cents a dozen, in cash ? Ans. 18 cts. a doz. 

10. If you sell 80 pounds of wool at 42 cents a pound, 

and in payment receive 1 barrel of mess beef for $15, and 
the balance in sugar at 8 cents a pound, what quantity of 
sugar do you get ? Ans. 232f lbs 

11. A. and B. barter ; A. has wood worth $5 a cord, 
and B. flour worth $6 a barrel, but for which he expects 
$7 a barrel in a barter trade ; at what price should B. 
barter his wood, and how much wood should he give for 
12 barrels of flour ? 

Ans. B. should barter his wood at $5.83^ a C. and give 
14 C. 3.2 ft. for 12 barrels of flour. 

12. A country trader agreed to exchange 553 lbs. of 

butter, worth 20 cents a lb., for 1 hhd. of white sugar, 
containing 7 cwt. 3 qrs. 16.8 lbs. ; how much a lb. was 
the sugar valued at ? Ans. 12£ cts. 


ASSESSMENT OF TAXES. 

Lesson 153. 

Definitions. Real Estate is composed of houses, lands, 
and other immovable property. 

Personal Estate is composed of money, cattle, horses, 
and other movable property. 

Poll tax is a tax of so much on every able bodied man. 

1. In the year 1835, the state tax of a certain town was 


What is real estate composed of? Personal estate ? 
What is a poll tax ? 




ASSESSMENT OF TAXES. 


183 


$2,368.07 ; the county tax was $939.81, and the town 
voted to raise $2,500 to defray town charges. The value 
of the real estate in town, together with that of all the per¬ 
sonal estate, was $354,000, and there were 663 polls, 
which are to pay £ part of the whole tax. What was A. ’s 
tax, whose real estate was valued at $4,373, personal es¬ 
tate at $813, and who was taxed for 2 polls ? Ans. $73.82. 

Explanation. What was the whole tax to be assessed 
on the polls and estates of the town ? What sum did the 
tax on the polls, and on each of the polls, amount to ? 
What was the remaining tax to be assessed on the estates ? 
As $354,000 paid the remaining tax, what did $1 pay ? 
What did the sum of A.’s real and personal estate pay ? 
What was his whole tax then, including the tax for 2 polls ? 

2. What was B.’s tax in the same town, the value of 
whose real estate amounted to $1,000, personal estate to 
$1,175, and who was taxed for 3 polls ? Ans. $34.12. 

Note. It frequently happens that a person is taxed for more property 
than belongs to him, in which case the assessors reduce his tax; more¬ 
over, many persons are unable to pay their taxes. The consequence is, 
if the exact amount of taxes to be raised in a town be assessed on the 
polls and estates, the sum actually paid will be less than the tax re¬ 
quired. To obviate this difficulty, the assessors usually add 2 or 3 per 
cent, of a tax to the tax, and consider the amount as the sum to be as¬ 
sessed ; by which means the sum of all the taxes paid, will generally 
exceed the taxes required. 

3. -The whole amount of the taxes to be paid in a cer¬ 
tain town, in the year 1830, was $1,286.40 to which the 
assessors added 2 per cent. ; the value of the real estate, 
together with that of all the personal estate, was $117,273, 
and there were 211 polls, the tax on each of which was 
fixed at $1.20. What was G.’s tax, who possessed real es¬ 
tate to the amount of $1,583, personal estate to the amount 
of $275, and who was taxed for 1 poll ? Ans. $17.98. 

Assessors usually put the tax paid by various sums, 
in a table, by means of which any person’s tax can be de¬ 
termined with great ease. On the next page is such a 
table, made from the last example as follows, 

$1,286.40 tax. 

_ .02 

25.7280 2 per cent. 

What frequently happens? What is the consequence ? How is this 
difficulty obviated ? 

What do assessors usually do? Explain how a table is made from 
the last example. 




184 


ASSESSMENT OF TAXES. 


To tax $1,286.40 
add 2 per cent. 25.728 


211 polls. $1,312.13 amount to be assessed. 

$1.20 tax on each poll. 253.20 tax paid by the polls. 

422 $1,058.93 tax to be assessed on 

211 the real and personal 

- estates. 

$253.20 tax paid by the polls. 

8 

117273 ) 1058.930 ( .0090296 tax on $1, about. 

1055 457 


347300 

234546 


1127540 

1055457 


720830 

703638 


TABLE. 


The tax paid by 
$ $ 

$ 


$ 

$ 


$ 

1 is .009. 

by 20 

is 

.181. 

by 200 

is 

1.806. 

2 “ .018. 

30 

( c 

.271. 

300 

cc 

2.709. 

3 “ .027. 

40 

it 

.361. 

400 

c c 

3.612. 

4 “ .036. 

50 

a 

.451. 

500 

c c 

4.515. 

5 “ .045. 

60 

tt 

.542. 

600 

c c 

5.418. 

6 “ .054. 

70 

a 

.632. 

700 

cc 

6.321. 

7 “ .063. 

80 

tt 

.722. 

800 

c c 

7.224. 

8 “ .072. 

90 

c t 

.813. 

900 

c c 

8.127. 

9 " .081. 

10 " .09. 

100 

et 

.903. 

1,000 

cc 

9.03. 


G.’s tax is found from the table as follows ; 
$1,583 G.’s real estate. 

275 G.’s personal estate. 


$1,858 whole estate. 













SIMPLE FELLOWSHIP. 


135 


By the table, the tax on $1,000, is $9.03. 
t( “ «« ann n no a 


it 

it 

“ on 

800, 

is 

7.224. 

it 

a 

" on 

50, 

is 

.451. 

it 

tt 

“ on 

8, 

is 

.072. 


16.78 

1.20 poll tax. 


$17.98 G.’s tax. Ans. 

Explanation . The manner of forming this table is very 
plain; for having obtained the tax on $1, we multiply it 
by each of the other sums to get the tax which they pay. 
The table can be enlarged by carrying it up to $10,000, 
and down to 1 cent, if we choose. 

4. In example 3, add 3 per cent, to the amount of the 
taxes, and then find the tax to be paid by G. ; supposing 
each poll to pay $1.20 as before. Ans. $18.18. 


FELLOWSHIP, 


OR COMPANY BUSINESS. 

Lesson 154. 

When two or more persons associate together for the 
purpose of trade, they are said to enter into partnership, 
and are called a company or firm. The amount of property 
that each partner puts into the firm, is called his capital, 
or stock in trade , and his share of the gain is called his 
dividend. 

SIMPLE FELLOWSHIP. 

In Simple Fellowship, the capitals of the different part¬ 
ners are employed during equal times. 


What is the manner of forming this table ? Can the table be enlarg¬ 
ed, and how ? 

When two or more persons associate together for the purpose ot 
trade, what are they said to do? What are they called ? What is the 
capita], or stock in trade of each partner ? His dividend ? 

How are the capitals of the different partners employed in Simple 
Fellowship ? 


16 * 






186 


SIMPLE FELLOWSHIP. 


1. A. and B. entered into partnership for the purpose of 
trade ; A. put in $1,000, and B. $1,250 ; after trading 1 
year, they found a gain had been made of $945 ; what was 
each one’s dividend ? Ans. A.’s dividend $420, B.’s $525. 

Explanation. What was the sum put in by both ? As 
his sum gained $945, what did $1 gain ? W hat did $1,000 
gain ? 1,250 gain ? 

To prove Simple Fellowship, 

Add together all the shares in the gain or loss , and the sum 
will evidently he equal to the whole gain or loss, if the work 
he done right. 

2. James Wallace, William Clark, and Samuel Shaw, 
formed a connection in business under the firm of James 
Wallace and Co. The capital put in by Wallace was 
$2,500, by Clark $2,200, and by Shaw $12,000. On set¬ 
tling their business, at the end of 18 months, they found 
that the profits amounted to $6,500 ; what was each one’s 
share of this sum ? 

Ans. Wallace’s share $973.05, Clark’s $856.29, Shaw’s 
$4,670.66. 

3. A. and B. formed a company for trade ; A. put in 

$950, and B. $800 ; after trading a short time they found 
a loss had been made of $300; what was each one’s share 
of the loss ? Ans. A.’s share $162.86, B.’s $137.14. 

How much of his capital did A. save ? How much of his 
capital did B. save ? 

4. A. and B. bought a quantity of land on speculation, 
B. paying as much as A. They cleared $1,540 by the 
speculation ; what was each one’s dividend of the gain ? 

Ans. A.’s $990, B.’s $550. 

5. A gentleman who had 1 son and 2 daughters, left by 
his will $5,000 to the son, $3,000 to the eldest, and $2,500 
to the youngest daughter. At his death it was found 
that the property remaining, after paying his debts, was 
$14,950; what part of this sum should each of the children 
take ? 

Ans. the son $7,119.05, the eldest daughter $4,271.43, 
and the youngest daughter $3,559.52. 

6. The yearly profits of a cotton factory, valued at 
$17,000, and owned in 80 shares, amounted to $2,825.16, 
$275.16 of which it was thought prudent to keep back, to 


To prove Simple Fellowship how do we proceed ? 



COMPOUND FELLOWSHIP. 


187 


meet contingent expenses ; the remainder being divided 
among the owners, what was the amount of the dividend 
paid to the owner of 2 shares ? Ans. $63.75. 

7. If the factory, instead of yielding an income, had cost 

the proprietors $1,000, what part of this sum would the 
owner of 2 shares have paid ? Ans. $25. 

8. 4 merchants, A. B. C. and D., bought a ship for 
$14,000, of which A. paid $5,000, B. $2,000, C. $3,000, 
and D. $4,000. During her first voyage she earned 
$5,600 ; what was each one’s share of the gain ? 

Ans. A.’s share $2,000, B.’s $800, C.’s $1,200, D.’s $1,600. 


COMPOUND FELLOWSHIP. 

Lesson 155. 

In Compound Fellowship the capitals of the different 
partners are employed during unequal times. 

1. A. and B. traded in company ; A. had a capital of 
$400, which was employed 6 months, and B. a capital of 
$450, which was employed 8 months. They gained $120; 
what was the share of each ? 

Ans. A.’s share $48, B.’s $72. 

Explanation. How many dollars should A. have em¬ 
ployed 1 month, to be equal to the use of $400, 6 months ? 
How many dollars should B. have employed 1 month, to 
be equal to the use of $450, 8 months ? The question now 
is evidently the same as if we were required to find the 
share of A. and B. in a gain of $120, if A. had put in 
$2,400, and B. $3,600, and had continued these sums in 
trade 1 month. 

Therefore, to perform an example in Compound Fellow¬ 
ship, 

Consider the product of each partner’s capital by the time 
it was continued in trade , as constituting his capital , and 
then proceed as in Simple Fellowship. 


How are the capitals of the different partners employed in Compound 
Fellowship ? 

How do we proceed to obtain the answer to example 1, lesson 155? 
How then do we perform an example in Compound Fellowship ? 




188 


COMPOUND FELLOWSHIP. 


To prove Compound Fellowship, 

Proceed as in Simple Fclloioskip. 

2. Charles Jones, Henry Adams, and John Stevens 
formed a company, under the firm of Jones, Adams, and 
Co., and commenced trade the first of June, on $2,000 
put in by Jones; the first of August, Adams put in $3,000, 
and the first of September, Stevens put in $4,000. At the 
end of the year their gains amounted to $1,500 ; what was 
each partner’s share ? 

Ans. Jones’s share $466.67, Adams’s $500, Stevens’s 
$533.33. 

3. A.'and B. formed a partnership the first of January, 
and put in $3,000 apiece. The first of April, A. put in 
$1,000 more, and the first of September, B. put in $500 
more. At the end of the year they found the whole of 
their gain to be $"2,000 ; what was each one’s dividend ? 

Ans. A.’s dividend $1,084.34, B.’s $915.66. 

4. A. and B. hired a pasture for $36 ; A. put in 6 cows 

for .5 of a year, and B. 4 cows for .25 of a year ; what 
should each one pay ? Ans. A. $27, B. $9. 

5. A. B. and C. entered into partnership. A. kept his 
capital in 1 year. B. put in f- as much as A., and em¬ 
ployed it 9 months. C. put in f as much as B., and em¬ 
ployed it 6 months. They gained $3,000 ; what was each 
one’s share of the profit ? 

Ans. A.’s share $1,610.74, B.’s $906.04, C.’s $483.22. 

Explanation. Consider that A. put in $1. 

6. A. and B. traded in company. A.’s capital was $5,000 
at the commencement, but at the end of 4 months he took 
out $3,000, and kept the remainder in the company 6 months 
longer; B.’s capital was $3,000 at the commencement, but 
at the end of 5 months he put in $4,000 more, and con¬ 
tinued the whole in the company 3 months longer. They 
found, on settlement, that a loss had been sustained of 
$1,800 ; what was each one’s share pf it ? 

Ans. A.’s share $847.06, B.’s $952.94. 


To prove Compound.Fellowship how do we proceed ? 



INSURANCE. 


189 


INSURANCE. 

Lesson 156 . 

Insurance is an agreement to pay the damages which 
vessels, goods, houses, See. may sustain from certain acci¬ 
dents, like the perils of the sea, fire, &c. 

The written promise to pay such damages is called a 
policy of insurance. 

The compensation paid to obtain insurance is called the 
premium. It is usually a certain per cent, of the sum 
insured. 

The insurer is often called the underwriter. 

Marine Insurance. When property is insured against 
loss or damage from the perils of the sea, the owner is paid 
on any loss, the same part of such loss as the sum insured 
is of the whole value of the property. 

If a person desires to be insured so as to lose nothing in 
case of a total destruction of the property, not even the pre¬ 
mium and other costs of insurance, he has the premium 
and other costs of insurance added to the value of the 
property, and insures the whole amount. Property so in¬ 
sured is said to be covered. 

Policies are usually so made that the insurer is not liable 
for any losses under five per cent. 

When property is greatly damaged, the insured can, if 
he pleases, abandon it as a total loss, in which case the 
underwriter must pay the sum insured, and take what is 
saved ; for instance, if goods be damaged more than half 
their value by the perils insured against, or if a ship be 
damaged by such perils so as to require* repairs exceeding 
half its original value, after deducting from the cost of 
such repairs for the superiority of the new work over the 
old, according to custom. However, when property is 


What is insurance ? 

What is called a policy of insurance ? 

What is called the premium r What is it usually ? 

What is the insurer often called ? 

When property is insured against loss or damage from the perils of 
the sea, what is the owner paid on any loss ? 

What if a person desires to be insured so as to lose nothing, in ca9e 
of a total destruction of the property, not even the premium and other 
costs of insurance ? What is said of property so insured ? 

What is said of losses under five per cent ? 

What if property be greatly damaged ? Give some instances P 




190 


INSURANCE. 


abandoned in this way, the underwriter can make repairs, 
if he pleases, and deliver it to the insured, obliging him to 
pay ^ of the cost of the repairs, for the superiority of the 
new work over the old. 

Fke Insurance. When property is insured against 
loss or damage from fire, the owner is reimbursed for any 
loss, unless it exceeds the sum insured. 

Lesson 157. 

1. A man had his house and furniture insured against 

loss or damage from fire during one year, at § of 1 per 
cent., what did the premium amount to, if the property was 
valued at $3,800 ? Ans. $25.33^. 

2. A merchant had $5,000 insured on his brig, valued 

at $8,000, during her voyage from New York to Havana. 
The premium was 2 per cent., for which he gave his note ; 
what must he receive from the underwriters if the vessel 
was damaged to the amount of $2,000, after deducting the 
amount of his note ? Ans. $1,150. 

3. If I have $5,000 insured on my store for one year, 

and obtain the policy for $37.50, what per cent, premium 
do I pay ? Ans. .75 per cent. 

4. What sum must be paid for insuring a ship worth 

$9,000 from Portland to Matanzas, at 3 per cent., from 
Matanzas to Canton, at 4 per cent., and from Canton to 
Portland, at 6 per cent. ? Ans. $1,170. 

5. If I have a vessel valued at $2,112, bound to Lisbon 
from Baltimore, what sum must I get insured to cover it, 
and what will be the premium at 2^ per cent., the commis¬ 
sion to agent for making insurance being £ per cent., and 
the commission for recovering loss, if any be sustained, 1 
per cent. 

OPERATION. 

Premium,.. 

Commission for making insurance, 

“ “ recovering loss, .. 

100 
4 

$2,112 then is 96 per cent, of the sum to be insured. 

What can the underwriter do when property is abandoned in this way ? 

When property is insured against loss or damage from fire, what if 
the owner paid for any loss ? 


2£ per cent. 
x i( 

j 2 n a 







INSURANCE. 


191 


$ 

96 ) 2112.00( 2200 sum to be insured to cover $2,112. 
192 ,02£ See Percentage, lesson 121. 

192 44 

192 11 


$55,000 premium, 2£ per cent, of $2,200. 

PROOF. 


From sum to be insured,.$2,200 

Deduct 2£ per cent., premium,.$55 


£ per cent., commis. for making ins. 11 
1 per cent., commis. for recov. loss, 22 

88 


Amount covered,.k.$2,112 

Therefore, to find the sum to be insured to cover a cer¬ 
tain amount, 

Find what per cent, the costs of insurance are , subtract 
this per cent, from, 100 per cent., and divide the amount to be 
covered by the remainder . 

Then, to find the amount of the premium, 

Multiply the sum'to be insured by the premium per cent. 

And to prove the work, 

Deduct the costs of insurance from the sum to be insured , 
and the remainder will be the amount covered. 

6. If you insure a ship worth $8,000, at 5 per cent., so 
as to cover her value during her passage from Manilla to 
Salem, what amount of premium must you pay ? 

Ans. $421.05 

7. What is the premium for insuring a quantity of goods 
worth $10,000, at 4 per cent. ; and what is the sum cover¬ 
ed by such an insurance ? 

Ans. the premium is $400, and the sum covered $9,600. 

8. What amount of premium, at 3 per cent., must be 
paid to cover $8,000 worth of goods during their transpor¬ 
tation from Boston to Tampico, the commission for making 


Explain how example 5, lesson 157, is performed. 

How do we find the sum to be insured to cover a certain amount? 
How then do we find the amount of the premium.? 

How do we prove the work.? 








192 


GENERAL AVERAGE. 


insurance being ^ per cent., and the commission for recov¬ 
ering loss £ per cent. ? Ans. $249.35. 

9. If $88.10 be the amount of the premium for insuring 
a quantity of goods during their transportation from New 
York to Bangor, at 1 per cent., what is the sum insured, 
and what is the sum covered ? 

Ans. the sum insured is $8,810, and the sum covered 
$8,721.90. 


GENERAL AVERAGE. 

Lesson 158. 

When a vessel is in danger, it is sometimes necessary 
to cut away the masts, spars, rigging, &.C., or to throw 
overboard a part of the cargo, in order to preserve the 
remaining property. When such a sacrifice is made, each 
of the owners of the ship and cargo suffers a portion of the 
loss, according to his part of the property, whether sacri¬ 
ficed or not. 

When a vessel is stranded, or meets with a like disaster 
through accident, and by extraordinary expense is got off, 
and enabled to pursue her voyage, the expense is appor¬ 
tioned on the owners of the ship and cargo. 

When a vessel is obliged, from the occurrence of extra¬ 
ordinary accidents, to enter a port to repair, the provisions 
and wages of the seamen during her stay, and indeed all 
the expenses attending the delay, except the cost of re¬ 
pairs, are likewise borne by apportionment, or general 
average. 

The damage a vessel or cargo may suffer from acci¬ 
dents, is borne by the owner of the property injured, or by 
the underwriter, if insured. Before making a general 
average, the amount of such damage must be deducted 


What is sometimes necessary when a vessel is in danger ? When 
such a sacrifice is made, who suffers the loss ? 

What is said of a vessel’s being stranded, and got off? 

When a vessel is obliged, from the occurrence of extraordinary acci¬ 
dents, to enter a port to repair, what is'said of the expenses attending 
the delay ? 

By whom is the damage a vessel or cargo may suffer from accidents 
borne ? What is done before making a general average ? 




GENERAL AVERAGE. 


193 


from the value of the injured property ; the costs of insur¬ 
ance must also be deducted from the value of the property 
insured ; likewise all charges that lessen the value of any 
property must be deducted from it. § of the money paid 
for freight must be added to the value of the ship ; it being 
supposed that £ of the amount of the freight is expended in 
wages and provisions for'the seamen. In New York, only 
£ of the freight is added to the value of the ship. 

When the masts, spars, rigging, 8tc. of a vessel are de¬ 
stroyed for the general good, the damage is reckoned § as 
much as the cost of repairing ; the new articles being sup¬ 
posed £ better than the old ones. 

The underwriter is liable for the expense contributed to 
a general average by insured property, even if the sum so 
contributed be less than five per cent, of the amount insur¬ 
ed, whatever the policy may say. 

1. A brig worth $9,016, covered by insurance at 2 per 
cent., and belonging to A. of Boston, was loaded with 
cloths and other goods for B., worth $15,000, insured at 3 
per cent., and with hardware for C., worth $5,000, not in¬ 
sured. The amount of the freight was $1,482. On the 
voyage the brig encountered a heavy gale, and the master 
was obliged to throw overboard a G f the goods' belonging 
to C., and to cut away spars and rigging which it cost 
$158.25 to repair. What is each owner’s portion of the 
loss ; how much must C. receive, and how much must A. 
pay out ? 


OPERATION. 

Ship,.....$9,016 

Deduct premium for insurance, 2 per cent.,. 184 

$8,832 

Add § of freight,.►. 988 

Ship, and freight, .$9,820 


When the masts, spars, rigging, &c. of a vessel are destroyed for 
the general good, how much is the damage reckoned ? 

What is said of the liability of the underwriter in a general average ? 
Explain how example 1, lesson 158, is performed. 


17 









194 


✓ 


GENERAL AVERAGE. 


B.’s goods,...$15,000 9,820 

Deduct premium for insurance, 3 per 

cent.,. 450 


14,550 14,550 

C.’s goods,. 5,000 


Value of property to bear the loss,.$29,370 

LOSS. 

f of C.’s goods,.$4,000 

Damage of ship § of $158.25,. 105.50 

Freight lost on goods thrown overboard, 300 


Whole loss,.... 4,405.50 

$ 


29370 ) 4405.50 ( .15 loss borne by each dollar of the 
2937 0 property. 


1468 50 
1468 50 


$9,820 multiplied by $.15 gives $1,473.00 A.’s loss. 

14,550 multiplied by .15 gives 2,182.50 B.’s loss. 

5,000 multiplied by .15 gives 750.00 C.’s loss. 


4,405.50 whole loss. 
$4,000 value of C.’s goods sacrificed. 

750 deduct C.’s share of the loss. 

3,250 sum that C. must receive. 

1,473 A.’s share of the loss. 

405.50 deduct A.’s property lost and sacrificed. 


1,067.50 sum that A. must pay out. 

Note. The insurers must ultimately pay what the brig contributes to 
the general average of the loss, and also B.’s share of the loss. As for 
C., not being insured, he must suffer his share of the loss, or $750. 

Lesson 159. 

1. Ship Triton, belonging to L. Murdock, went ashore 9 
miles from New York, during a storm, but by great exer- 


Who must ultimately pay what the brig contributes to the general 
average ? B.’s share of the loss ? C.’s share of the loss ? 


















GENERAL AVERAGE. 


195 


tions, after throwing over part of her cargo, she was got 
off, and towed up to the city by a steamboat. The follow¬ 
ing losses and expense are to be apportioned among the 
owners. 

Loss of cable and anchor, with some spars and rig¬ 


ging ; § of the cost of new articles,.$425 

Goods of John Williams, thrown overboard,. 2420 

Freight lost on the above goods,. 121 

Expense of steamboat 3 hours, at $70 an hour,. 210 

Protest,. 16 

Adjusting average,. 45 


Loss and expense to be apportioned,.3,237 


Property to pay the preceding, the premium for insur¬ 
ance having been deducted ; 

Ship,.$15,000 


£ of freight, . .*.. 



16,000 ship and freight. 

Goods of John Williams, including 

those lost,. 

.8,250 

Goods of Daniel Drake, ... 

.15,000 

Goods of T. Jones, ........ 


Goods of S. Hyde, . 


Total,... 



What will be each owner’s share of the loss and ex¬ 
pense ? 

Ans. L. Murdock’s $849.05, John Williams’s $437.79, 
Daniel Drake’s $795.98, T. Jones’s $583.72, S. Hyde’s 
$570.46. 

2. Barque Berosus, from New Orleans to Philadelphia, 
met with much bad weather, sprung her foremast, had 
several sails carried away, and was otherwise so much 
damaged as to render it prudent to put into Norfolk to re¬ 
pair. All the expenses attending the delay, including pi¬ 
lotage, dockage, protest, commission, provision and wages 
of the seamen, were $345. The vessel was valued, in 
New Orleans, at $10,000, of which the sum of $7,000 was 
insured at 3 per cent., and had received damage to the 
amount of $1,140. The freight amounted to $1,050. All 
of the cargo was insured, at 2£ per cent., and consisted 



















196 


ALLIGATION MEDIAL. 


of goods for A., valued at $9,000 in New Orleans, but 
which had been damaged to the amount of $300 ; goods 
for B., worth $5,000; goods for C., valued in New Orleans 
at $12,800, but which had been damaged to the amount of 
$587.50, and goods for D., worth $8,750. How much of 
the expense must the owner of the barque, and each owner 
of the cargo pay ? 

Ans. the owner of the barque must pay $74.80, A. 
$67.80, B. $39, C. $95.15, D. $68.25. 


ALLIGATION, 

OR MIXTURE. 

Lesson 160. 

Alligation is mostly used in mixing the precious metals. 

Jewellers divide an ounce of gold into 24 parts, called 
carats. They express the quality, fineness, or purity of 
gold in carats. Thus a piece of gold 24 carats fine, is 
pure, or fine gold ; a piece 22 carats fine, is of such a 
quality that an ounce of it contains 22 carats of pure or 
fine gold, and 2 carats of some base metal, like copper, 
called alloy. To express the quality, fineness, or purity of 
silver, jewellers name the quantity of pure or fine silver 
contained in 12 ounces of any kind. Thus, a piece of sil¬ 
ver 12 ounces fine, is pure or fine, silver ; a piece 11 oz. 
3 pwts. 17 grs. fine, is of such a quality that 12 ounces of 
it contains 11 oz. 3 pwts. 17 grs. of fine silver, and the 
rest alloy. Alloy is considered of no value. 


ALLIGATION MEDIAL. 

A method of finding the average price or quality of sev¬ 
eral ingredients mixed together. 


What is Alligation mostly used in ? 

How do jewellers divide an ounce of gold ? What do they express 
in carats P Give some examples. How do jewellers express the quali¬ 
ty, fineness, or purity of silver ? Give some examples. How is alloy 
considered ? 

Wljat is Alligation Medial ? 





ALLIGATION ALTERNATE. 


197 


1. A farmer mixed 4 bushels of wheat worth $.80 a 
bushel, with 6 bushels worth $1 a bushel, 8 bushels worth 
$1.30 a bushel, and 10 bushels worth $1.40 a bushel; 
what was the value of a bushel of the mixture ? Ans.$1.20. 

Explanation. What was the value of all the wheat 
mixed ? How many bushels were there ? Then what was 
1 bushel of the mixture worth ? 

2. A grocer mixed 200 pounds of sugar worth 6 cents a 

pound, with 400 pounds worth 8 cents a pound, and 500 
pounds worth 10 cents a pound ; what were 100 pounds of 
the mixture worth ? Ans. $8.55. 

3. What was 1 pound of the mixture worth ? 

Ans. 8.55 cts. or 8£ cts., about. 

4. A jeweller melted together 9 ounces of gold 18 carats 

fine, 10 ounces 19 carats fine, and 7 ounces of pure gold, 
which is 24 carats fine ; how many carats fine was the 
mixture ? Ans. 20. 

5. If 11 oz. of gold 18 carats fine, be melted with 2 lbs. 
10 oz. 20 carats fine, 8 oz. of pure gold, and 3 oz. of cop¬ 
per alloy, how many carats fine will the mixture be ? 

Ans. 19^-. 

6. A grocer bought 60 gallons of rum at $1 a gallon, 
mixed 15 gallons of water with it, and sold the mixture so 
as to gain 15 per cent.; at what price did he sell a gallon ? 

Ans. 92 cts. 

7. A silversmith melted together 10 oz. of silver contain- 
ing T \y alloy, 9 oz. containing alloy, and 5 oz. contain- 
ing alloy ; what proportion of the mixture was alloy ? 

Ans. T ^-. 

8. How would a jeweller express the fineness of the 
mixture ? 

9. If 5 lbs. 3 oz. of silver 10 oz. 1 pwt. fine, be melted 
with 3 lbs. of pure silver, and 9 oz. of copper, of what fine¬ 
ness will the mixture be ? Ans. 9 oz. 17 pwts. 6 grs., fine. 


ALLIGATION ALTERNATE. 

A method of finding how much of each ingredient must 
be taken to form a mixture of a certain price or quality. 


What is Alligation Alternate ? 

17* 




198 


ALLIGATION ALTERNATE. 


Lesson 161. 


1. A grocer has some sugar worth 7 cents a pound, and 
some worth 12 cents ; in what proportion must the two 
kinds be mixed so that the compound may be worth 10 
cents a pound ? 

OPERATION. 

10 12 

7 10 


3 2 

3 lbs. at 12 cts., and 
lbs. at 7 cts. Ans. 


Explanation. We find that the value 
of 1 lb. of the mixture exceeds thp 
value of 1 lb. of the cheapest sugar 
by 3 cents, and that the value of 1 lb. 
of the dearest sugar exceeds the value 
of 1 lb. of the mixture by 2 cents. 
Now if we multiply the 3 cents by 2, 
and the 2 cents by 3, the products will be alike, and each 
will be 6. The value of 2 lbs. of the cheapest sugar, then, 
is 6 cents less than the value of 2 lbs. of the mixture, and 
the value of 3 lbs. of the dearest sugar is 6 cents more than 
the value of 3 lbs. of the mixture ; so if we add 2 lbs. of 
the sugar worth 7 cents a lb., to 3 lbs. of that worth 12 
cents, we get 5 lbs. of the same value a pound as the mix¬ 
ture required. 


Therefore, to obtain the proportion in which two ingre¬ 
dients of given values are to be mixed to produce a com¬ 
pound of a certain price or quality, 

Take the difference between the values of the cheapest in¬ 
gredient and the compound to express the proportion of the 
dearest ingredient , and the, difference between the values of 
the dearest ingredient and compound to express the proportion 
of the cheapest ingredient. 

2. A grocer had some tea worth 45 cents a pound, and 
some worth 62 cents a pound ; in what proportion must 
they be mixed so that the compound may be worth 50 cts. 
a pound ? Ans. 12 at 45 cts., and 5 at 62 cts. 

Note. The numbers expressing the proportion may be considered as 
meaning pounds, ounces, or any other denomination; also both num¬ 
bers may be multiplied, or both divided by the same number, and the 
two products, or the two quotients, will evidently bear the same propor¬ 
tion to each other as the numbers do. 


3. A watchmaker has some gold 20 carats fine, and some 


Explain how example 1, lesson 161, is performed 
How do we obtain the proportion in which two ingredients of given 
values are to be mixed to produce a compound of a certain price or 
quality ? 

What is observed of the numbers expressing the proportion P 



ALLIGATION ALTERNATE. 199 

17 carats fine ; how many ounces of each kind must be 
melted so that the mixture may be 19 carats fine ? 

Ans. 2 oz. 20 carats fine, and 1 oz. 17 carats fine. 

4. A silversmith has some silver £ alloy, and some 

alloy ; in what proportion must he mix the two kinds to 

produce a compound ^ alloy ? 

Ans. in the proportion of to or °f 2 to 5. 

5. A grocer has some rum worth $2 a gallon ; in what 

proportion must this rum, and water worth $0 a gallon, be 
mixed, to produce a liquor worth a gallon ? 

Ans. 1£ of rum to £ of water, or 3 of rum to 1 of water. 

6. If you have some gold 23 carats fine, in what propor¬ 

tion must it be melted with alloy so that the mixture may 
be 20 carats fine ? Ans. 20 of gold to 3 of alloy. 

7. If you have some silver 5 per cent, copper, and some 
23 per cent, copper, how many ounces of each must be 
melted together so that the mixture may be 10 per cent, 
copper ? 

Ans. 13 oz. of 5 per cent, copper, and 5 oz. of 23 per 
cent, copper. 


Lesson 162. 


1. A miller has several kinds of corn worth 60, 70, 80, 
110, and 120 cents a bushel ; in what proportions must 
these kinds be mixed so that the compound may be worth 
100 cents a bushel ? 

OPERATION. 

100 120 

60 100 


40 20 

40 at 120 cents, and 20 at 60 cents. 
100 120 

70 100 


30 20 

30 at 120 cents, and 20 at 70 cents. 
100 110 

80 100 


> Answer. 


20 10 

20 at 110 cents, and 10 at 80 cents. 





ALLIGATION ALTERNATE. 


200 

Explanation. We find how a compound worth 100 cents 
a bushel, can be formed by mixing the corn worth 60 cents 
with that worth 120 cents, the corn worth 70 cents with 
that worth 120 cents, and the corn worth 80 cents with 
that worth 110 cents. These three compounds, each worth 
100 cents a bushel, when mixed together give another com¬ 
pound worth 100 cents a bushel. 

Now if the miller chooses, he can mix 40 quarts at 120 
cents with 20 quarts at 60 cents, 30 pecks at 120 cents 
with 20 pecks at 70 cents, and 20 bushels at 110 cents with 
10 bushels at 80 cents ; in fact, he may consider the num¬ 
bers expressing the proportions in which 2 of the kinds are 
to be mixed, as representing any measure that he pleases. 
It is usual, however, to consider all of the numbers ex¬ 
pressing the proportions, as representing the same meas¬ 
ures. If all the preceding numbers are considered as 
representing bushels, the 40 bushels at 120 cents, and the 
30 bushels at 120 cents are added together, and the quan¬ 
tity of each kind to be mixed are 20 bushels at 60 cents, 
20 bushels at 70 cents, 10 bushels at 80 cents, 20 bushels 
at 110 cents, and 70 bushels at 120 cents. We may mul¬ 
tiply or divide the numbers expressing the proportions in 
which 2 or all of the kinds are to be mixed by any number, 
and we shall still have the true proportions. Moreover, 
we can combine the several kinds by 2 and 2 in many ways 
different from the preceding, and thereby obtain many dif¬ 
ferent, but correct answers ; for instance, we can find in 
what proportions to mix the corn at 60 cents with that at 
110 cents, the corn at 70 cents with that at 110 cents, and 
the corn at 80 cents with that at 120 cents. The only pre¬ 
caution to be observed in combining 2 kinds, is to take one 
kind cheaper , and the other clearer than the required com¬ 
pound. 

Therefore, when several ingredients of different values 
are to be mixed, so as to form a compound of a certain 
value, 


Explain how example 1, lesson 162, is performed. 

In what way can the miller mix the different kinds ? What in fact 
may he do ? What, however, is usual ? 

What if all the preceding numbers are considered as representing 
bushels? What if we multiply or divide these numbers? What is 
said of getting different answers ? Give an example. What is the only 
precaution to be observed ? 



ALLIGATION ALTERNATE. 


201 


Take one ingredient lohose value is less, and another whose 
value is more than the compound required, and find the pro¬ 
portion in which they are to be mixed to produce the compound ; 
combine the other ingredients by 2 and 2 in the same manner 
and mix the whole in the proportions obtained. 

2. A goldsmith has several kinds of gold, 16, 18, 19, and 
23 carats fine; in what proportions must the different kinds 
be mixed so as to produce a compound 20 carats fine ? 

Ans. 3 of 16 carats and 4 of 23, 3 of 18 and 2 of 23, 3 
of 19 and 1 of 23. 

3. A silversmith has several kinds of silver, containing 
£> aQ d alloy ; what proportions of the different 
kinds must he take so as to form a mixture containing £ 
alloy ? 

Ans. 4 of £ alloy and 5 of -^y, 10 of £ and 5 of 1 ^-. 

4. If you have pure gold, which is 24 carats fine, and 
some 17 and 20 carats fine, how must you mix the dif¬ 
ferent kinds so as to make a compound 22 carats fine ? 

Ans. 5 of 24 carats with 2 of 17, 2 of 24 with 2 of 20. 

5. A trader has some spirit worth $2.80 a gallon, some 
worth $2.10, some worth $1.60, and some worth $1.50 ; 
how many gallons must he take of each kind so as to make 
a compound worth $1.75 a gallon ? 

Ans. 15 at $2.80 and 105 at $1.60, 25 at $2.10 and 35 
at $1.50. 

Lesson 163. 

1. A farmer has 30 bushels of rye worth 80 cents a 
bushel, and wishes to mix with it some worth 90 cents, 
and some worth 112 cents a bushel ; what quantity must 
he take of that worth 90 cents, and of that worth 112 cents, 
to make with the 30 bushels a compound worth $1 a bushel ? 

Ans. 30 bushels at 90 cts., and 75 at 112 cts. 

Explanation. How many bushels at 80 cents, at 90 
cents, and at 112 cents, must be taken to make a com¬ 
pound worth $1 a bushel ? How many bushels of the two 
last mentioned kinds, are to be mixed with 1 bushel of the 
first, to make such a compound ? How many bushels of 
the two last mentioned kinds are to be mixed with 30 bush¬ 
els of the first to make such a compound ? 


What course do we take, when several ingredients of different values 
are to be mixed, so as to form a compound of a certain value ? 



202 


ALLIGATION ALTERNATE. 


2. I ha ye 8 ounces of gold 23£ carats fine, that I wish 
to melt with some 18, some 19, and some 21 carats fine, 
so that the mixture may be 20 carats fine ; how many 
ounces of each kind must I use ? 

Ans. 14 oz. 18 carats, 4 oz. 19 carats, and 4 oz. 21 carats. 

3. If you have some wine worth $3 a gallon, and some 
worth $2, what quantity of each sort must you mix with 
10 gallons of water worth $0 a gallon, so that the com¬ 
pound may be worth $1£ a gallon ? 

Ans. 7^ gallons of each sort. 

4. A grocer has 15 pounds of sugar worth 10 cents a 
pound, and 5 pounds worth 6 cents a pound ; what quan¬ 
tity of some worth 11 and some worth 15 cents a pound, 
must he mix with the two quantities, so that the compound 
may be worth 10 cents a pound ? 

Ans. 3^ lbs. at 11 cts., and 3^ lbs. at 15 cts. 

Explanation. The two first named kinds being united 
together, what is the value of a pound of the mixture ? 

5. A jeweller has 6 ounces of gold 16 carats fine, and 

10 ounces 22 carats fine, which he wishes to melt with 
some 21 carats fine, so that the mixture may be 20 carats 
fine ; how many ounces of the gold 21 carats fine must he 
use ? Ans. 4 oz. 

6. A man has 8 ounces of silver 10 oz. fine, 10 ounces 
9 oz. 14 pwts. fine, and 2 ounces 11 oz. 10 pwts. fine, 
which he wishes to melt with some 11 oz. 4 pwts. fine, and 
some pure silver, so that the mixture may be 11 oz. fine ; 
what quantity of the two last kinds must he take ? 

Ans. 16§ oz. 11 oz. 4 pwts. fine, and 16§ oz. of pure silver. 

7. A farmer has oats worth 40, 60, and 70 cents a bushel; 
what quantity of each of these kinds must he take to make 
40 bushels worth 50 Cents a bushel ? 

Ans. 24 bu. at 40 cts., 8 at 60 cts., and 8 at 70 cts. 

Explanation. How many bushels of each kind must be 
mixed in order to obtain a compound worth 50 cents a 
bushel ? How much of each kind must be taken to make 
1 bushel of such a compound ? How much of each kind 
must be taken to make 40 bushels of such a compound ? 

8. A jeweller wishes to make a piece of gold to contain 
12 ounces 21 carats fine, by melting together 3 kinds, 18£, 
20, and 23 carats fine ; how many ounces of each of these 
kinds must be taken ? 

Ans. 3| oz. 184, 3| oz. 20, and 5£ oz. 23 carats. 


PROMISCUOUS QUESTIONS. 


203 


9. A grocer has different kinds of wine, worth $3, $2, 
and $1£ a gallon ; how much of each of these kinds must 
be mixed with water so that the compound may till a 20 
gallon keg, and be worth $1£ a gallon ? 

Ans. 10 gals, at $3, If gal. at $2, If gal. at $1£, and 
7f gals, of water. 


PROMISCUOUS QUESTIONS 


IN 

RULE OF THREE, FELLOWSHIP, INSURANCE, &c. 
Lesson 164. 

To be performed in the mind. 

1. How much sugar at 8 cents a pound, can you obtain 
in exchange for 6 lbs. of wool, at 41 cents a pound ? 

2. What must I pay to get my house and furniture in¬ 
sured at 14 per cent., their value being estimated at 
$2,500 ? 

3. A schooner worth $4,000, was loaded with goods 
worth $4,000 by a merchant ; in a storm it became neces¬ 
sary to throw the goods overboard to preserve the vessel ; 
what sum must the owner of the vessel pay to the merchant 
on account of his loss ? 

4. A man paid $20 for 25 casks of lime ; what sum 
would 31 casks have cost him ? 

5. A. can finish a piece of work in 2 hours, and B. in 3 
hours ; how long will it take both to finish it ? 

6. If you mix 4 bushels of oats worth 50 cents a bushel 
with 6 bushels worth 60 cents a bushel, what will be the 
value of the compound ? 

7. How many gallons of cider worth 12 cents a gallon 
should be mixed with some worth 7 cents, so that the com¬ 
pound may be worth 10 cents a gallon ? 

8. A. owns | of a brig, B. f, and C. if she earns 
$1,000, what will be each one’s share of the profit ? 

9. If you commence trade the 1st of January on $2,000, 
and the 1st of July are joined by a partner with $1,000, 
what will be your share of $1,200 profit, found to have 
been gained at the end of the year ? 



204 


PROMISCUOUS QUESTIONS. 


10. What quantity of honey at 12 cents a pound must a 
man give for an axe worth $1.50, and 10 pounds of cheese 
worth 9 cents a pound. 

11. If 6 men consume 1 bl. of flour in 20 days, how long 
will 2 bis. last 1 man ? 

12. If 12 bushels of wheat be worth $25, what are 4 
bushels worth ? 

Lesson 165. 

1. I wish to cover the value of my brig, worth at 
$5,000, during a voyage from New York to Smyrna, the 
premium for insurance being 4 per cent. ; what amount 
must I insure, and what premium must I pay ? 

Ans. I must insure $5,208.33L and pay $208.33£ premium. 

2. A man paid $3.90 for the use of $80, 9 months ; how 

much should he pay for the use of $375, 11 months, at the 
same rate ? Ans. $22.34. 

3. A lumber dealer sold a lot of 80 thousand pine shin¬ 

gles for $1350 ; what should he sell 33 thousand for, at 
the same rate ? Ans. $144.37^. 

4. 7 men can remove 896 cubic yards of gravel in 8 

days ; how many men will it take to do the same labor in 
14 days ? Ans. 4. 

5. If a merchant barters 80 lbs. of rice, worth 4 cts. a 

pound, at 5 cts., what should corn worth $1 a bushel be 
bartered at in exchange ? Ans. at $1.25 

6. If 6§- yds. of cloth, If yd. wide, cost $25.50, what 
should 11 yds. of cloth of the same quality, 4 wide, cost ? 

Ans. $15.71. 

7. A. owns T 6 <y of a vessel, and B. the other ; what is 
each one’s share of the earnings for one year, amounting 
to $1,859 ? 

Ans. A.’s share is $1,115.40, and B.’s, $743.60. 

8. A. can do a piece of work in 6 days, and B. in 9 
days ; how long will it take them both to perform it ? 

Ans. 3§ days. 

9. If I lend you $2,000 for 1 year, how long should you 
allow me the use of $500 to reciprocate the favor ? 

Ans. 4 years. 

10. A man gave $1.50 for of a cord of wood ; what 
should he pay for 3£ cords, at the same rate ? 

Ans. $14.25. 


PROMISCUOUS QUESTIONS. 


205 


Lesson 166 . 

1. If you give $6,627 for 85 T. 6 cwt. 3 qrs. of Ameri¬ 

can bar iron, what quantity can you get for $453.33, at the 
same rate ? Ans. 5 T. 16f cwt., about. 

2. All the taxes to be paid in a certain town, in 1835, 

amounted to $4,000, which the assessors increased 2 per 
cent. ; all the real and personal estate in the town was 
valued at $200,000 ; there were 180 polls, the tax on each 
of which was fixed at $2 ; what did A.’s tax amount to, his 
real estate being valued at $3,000, personal estate at $185, 
with 1 poll ? Ans. $61.24. 

3. A cistern containing 1,000 gallons was emptied by 2 

cocks in 5 hours ; how long will it take 3 such cocks to 
empty it ? Ans. 3 h. 20 m. 

4. If you own £ of a tract of land containing 840 acres, 
and sell of your share for $656.25, how much should you 
ask for 30 acres of the remainder, at the same rate ? 

Ans. $187.50. 

5. A jeweller has gold 18, 19, and 22 carats fine ; what- 
quantity of each must be taken to make 1 lb. 20 carats fine ? 

Ans. f lb. 18 carats, f lb. 19 carats, and $ lb. 22 carats. 

6. If 2 lbs. 3 oz. of silver 11 oz. fine, be melted with 1 
oz. of copper, and 1 lb. of silver 10 oz. 10 pwts. fine, what 
will be the fineness of the mixture ? 

Ans. 10 oz. 11 pwts. 12 grs. fine. 

7. A., B. and C., traded in company ; A. put in $1,000, 
and employed it 1 year ; B. put in $4,000, and employed 
it 8 months, and C. put in $2,500, and employed it 10 
months ; they gained $2,000 ; what was each one’s share ? 

Ans. A.’s share $347.82, B.’s $927.54, C.’s $724.64. 

8. How much must I insure on my house to cover its 

value, it being worth $600, and the premium for insurance 
being 2£ per cent. ? Ans. $615.38. 


18 


MENSURATION. 


3J6 


MENSURATION. 

Lesson 167. 

Definitions. A corner is called an angle, and a square 
corner a right angle. An angle greater or less than a right 
angle, is called an oblique angle. 

Figure 3. 

3 feet long. By examining figure 3, we find as we 

1 did in Compound Numbers, that an ob¬ 
long square 1 ft. wide and 3 ft. long, 
contains 3 sq. ft.; that an oblong square 

2 ft. wide and 3 ft. long, contains 2 
times 3, or 6 sq. ft., and that a square 

3 ft. long and 3 ft. wide, contains 3 
times 3, or 9 sq. ft. 

Therefore, to get the surface of an oblong square, or of 
a square, 

Multiply the length by the breadth. 

A four sided figure, having 
its opposite sides equal and 
parallel, and its angles oblique, 
like figure 4, is called a paral¬ 
lelogram. 

If we cut a corner off from the left of figure 4, by dot¬ 
ting a line down perpendicularly to the lower side, and add 
it on at the right, we make an oblong square just as large, 
and of just the same length and breadth as the parallelo¬ 
gram. The product of this length and breadth is the sur¬ 
face of the oblong square, and also of the parallelogram, 
since they are equal. 

Therefore, to get the surface of a parallelogram, 

Multiply the length by the breadth. 

What is a corner called ? A square corner ? What is called an ob¬ 
lique angle ? 

What do we find by examining figure 3, lesson 167 ? 

How do we get the surface of an oblong square, or of a square ? 

What is called a parallelogram ? 

How do we proceed to find a way to obtain the surface of a parallel¬ 
ogram ? 

How do we get the surface of a parallelogram ? 


Figure 4. 






CO 












MENSURATION. 


2(y: 


A three sided figure, like 
figure 5, is called a triangle. 
If we dot a line down from a 
corner, or angle of the trian¬ 
gle, perpendicularly to the oppo¬ 
site side, the dotted line is call¬ 
ed the altitude of the triangle, and this opposite side is 
called the base. 

If we now dot lines parallel to the base and one of the 
sides, as in figure 5, we complete a parallelogram evidently 
twice as large as the triangle ; now the surface of the par¬ 
allelogram is equal to the product of its length by its 
breadth, the length being the same as the base of the tri¬ 
angle, and the breadth the same as its altitude. 

Therefore, to get the surface of a triangle, 

Take half the product of the base by the altitude. 

A four sided figure, having two sides 
parallel but unequal, like figure 6, is 
called a trapezoid. 

If we divide the trapezoid into 2 tri 
angles, the surface of one will be 
equal to half the product of its base, or the lower side of 
the trapezoid, by its altitude, or the distance between the 
parallel sides ; the surface of the other will be equal to 
half the product of its base, or the upper side of the trape¬ 
zoid, by its altitude, or the distance between the parallel 
sides. 

Therefore, to get the surface of a trapezoid, 

Take half the product of the sum of the two parallel sides 
by the distance between them. 


What is called a triangle ? What is called the altitude of a triangle ? 
The base ? 

How do we proceed to find a way to obtain the surface of a triangle? 
How do we get the surface of a triangle ? 

What is called a trapezoid ? 

How do we proceed to find a way to obtain the surface of a trapezoid ? 
How do we get the surface of a trapezoid P 



Figure 5. 



Base 







208 


MENSURATION. 


Lesson 168. 

If we have a long irregular piece, 
like figure 7, we can measure the 
breadth in various places, thereby di¬ 
viding it into trapezoids. 

Now to get the surface of the whole 
figure, 

Find the surface of the trapezoids separately , and add them 
together. 

We also can get the surface if we 

Multiply the average breadth by the length. 

To find the average breadth, measure the breadth at 
each end, and in several places, at equal distances apart, 
then take half the sum of the breadths at the two ends, 
add it to the sum of the intermediate breadths, and divide 
the result by the number of trapezoids. This way is cor¬ 
rect ; for to find the average breadth of each trapezoid, we 
take half the sum of its two parallel sides ; so the sum of 
all these average breadths contains half the sum of the first 
and last parallel sides, and the sum of all the intermediate 
ones ; the sum of the average breadths of all the trape¬ 
zoids divided by the number of trapezoids evidently gives 
the average breadth of the whole piece. 

Note. The usual way of finding the average breadth of a long irregu¬ 
lar piece, is to take the breadth at each end, and in several other places, 
at equal distances apart, and divide the sum of the breadths by their 
number. This course is incorrect, and absurd, but there are many other 
similar mistakes made in averaging ; thus, if the temperature is 30 de¬ 
grees at 6 o’clock in the morning, 60 degrees at noon, and 48 at 6 
o’clock in the evening, many persons add 30, 60, and 48 together, and di¬ 
vide by 3 to get the average temperature of the day ; on the contrary, 
we must add half of 30 and 48 to 60, and divide by 2, the number of in¬ 
tervals. If 20 cubic ft. of water run over a dam in a second, at 6 


How do we proceed to get the surface of an oblong irregular piece ? 
How also can we get the surface ? 

How do we find the average breadth ? Why is this way correct? 

What is the usual way of finding the average breadth of a long irregu¬ 
lar piece ? What is said of this course ? If the temperature be 30 de¬ 
grees at 6 o’clock in the morning, 60 degrees at noon, and 48 degrees at 
6 o’clock in the evening, what is the usual, and what is the correct way 
to get the average temperature of the day ? 

If 20 cubic ft. of water run over a dam in a second, at 6 o’clock in the 
morning, 16 cubic ft. at 10 o’clock, 14 at 2 o’clock, and 18 at 6 o’clock 
in the evening, what is the usual, and what is the correct way to get 
the average number of cubic ft. a second ? 


Figure 7. 








MENSURATION. 


209 


o’clock in the morning, 1G cubic ft. at 10 o’clock, 14 cubic ft. at 2 o’clock, 
and 18 cubic ft. at 6 o’clock in the evening, many persons add 20, 16, 
14, and 18 together, and divide by 4 to get the average number of cubic 
ft. passing over the dam in a second; on the contrary, we must add 
half of 20 and 18, to 16 and 14, and divide by 3, the number of intervals. 
There are many other cases where the learner will see the propriety of 
averaging in a similar manner. 

To find the surface of an irregular figure of 4 sides, or 
of a figure of 5, 6, 7, &c., sides, 

Divide it into triangles , and the sum of their surfaces will 
be the surface of the figure. 

The curved line that bounds a 
circle is called the circumference. A 
line drawn through the centre of a 
circle, touching the circumference 
at both ends, like the line A B, is 
called the diameter. Half the di¬ 
ameter is called the radius. 

The circumference of a circle has been found by calcu¬ 
lation to be about 3.14159 times the diameter ; for rough 
calculations it may be reckoned 3 times the diameter. 

Let us imagine a great number of triangles, very small 
indeed, to be formed in a circle, with their tops in the cen¬ 
tre, and having for bases portions of the circumference so 
small that they may be considered straight lines. The 
surface of each triangle is equal to half the product of its 
base by its altitude, the altitude being the radius ; the sur¬ 
face of all these triangles, then, or 

The surface of the whole circle, is equal to 

Half the product of the circumference by the radius. 

Lesson 169. 

1. How many square feet are there in a house lot 51 ft. 



How do we find the surface of an irregular figure of 4 sides, or of a 
figure of 5, 6, 7, &c. sides ? 

What is called the circumference of a circle ? Diameter? Radius? 
The circumference has been found by calculation to be how many 
times the diameter? For rough calculations how may it be reckoned ? 
How do we proceed to find a method of obtaining the surface of a circle ? 
What is the surface of a circle equal to ? 

18* 


I 






210 


MENSURATION. 


9 in. long, and 32 ft. 6 in. wide, and what will it cost at 
$.55 a sq. ft. ? 

Ans. 1,681.875 sq. ft., and it will cost $925.03. 

Note. See note after example 6, lesson 107, in Reduction of Com¬ 
pound Numbers. 

2. How many acres are there in a piece of ground 60 

rods square ? Ans. 22£. 

3. How large is the surface of one side of a board 18.67 

ft. long, and 1.25 ft. wide ? Ans. 23£ sq. ft., about. 

4. How many square yards are there in a piece of cloth 
27f- yards long, and £ of a yard wide ? Ans. 24.28, about. 

5. How much will it cost to plaster the ceiling of a room 
19 ft. 6 in. long, and 15 ft. wide, at 10 cts. a sq. yd. r 

Ans. $3.25. 

6. There is a house that has 12 windows 5 ft. 9 in. high, 

and 3 ft. 3 in. wide, and 7 windows 5 ft. 3 in. high, and 3 
ft. 2 in. wide ; what will the glazing of these windows cost 
at $.20 a sq. ft.? Ans. $68.12^. 

7. How much must I give for paving a yard 60 ft. 4 in. 
long, and 57 ft. wide, at $.40 for every 100 sq. ft. ? 

Ans. $13.76. 

8. What is the surface «f a field in the form of a paral¬ 

lelogram 16 rods 9 ft. long, and 5 rods wide, measuring 
square across ? Ans. 22,522£ sq. ft. 

9. A man bought a house lot in a triangular shape, the 

altitude being 40^ ft., and the base 73f ft. ; how much did 
it cost at 30 cents a sq. ft. ? - Ans. $447.52. 

10. What is the surface of a trapezoid, one of the paral¬ 

lel sides being 35.6 ft., the other 91.4 ft., and the distance 
between them 10 ft. ? Ans. 635 sq. ft. 

11. If a piece of land be 40 ft. long, and measuring the 

breadths at every 10 ft. you find them to be as follows ; 
8 ft., 5 ft., 2ft., 11 ft., and 13 ft., what is the surface of 
the whole piece ? . Ans. 285 sq. ft. 

12. What is the surface of a circular fish pond 50 ft. in 

diameter ? Ans. 1,963.49 sq. ft., about. 

Explanation. What is the circumference of the pond ? 

Lesson 170. 

1. W r hat is the surface of a semicircle, or half a circle, 
like half of figure 8, lesson 168, the diameter of the circle 
being 12 ft. ? Ans. 56.549 sq. ft., nearly. 


MENSURATION. 211 

2. What is the surface of a quadrant, or quarter of a 
circle, the diameter of the circle being 5£ rods ? 

Ans. 5.94 sq. rods, nearly. 

3. What is the diameter of a circle in feet and decimals, 
the circumference being 18 ft. 9 in. ? Ans. 5.968 ft., about. 

4. The surface of a field in the shape of an oblong 

square, is 1 acre, and its length is 16 rods ; what is its 
breadth ? Ans. 10 rods. 

5. A parallelogram containing 225 sq. ft. is 12 ft. 6 in. 

wide ; how long is it ? Ans. 18 ft. 

6. A triangle containing 2 acres, has a base 20 rods 

long what is its altitude ? Ans. 32 rods. 

7. How many square yards of papering are there in a 

room 18 ft. long, 16 ft. wide, and 10 ft. high, if we dedtict 
the space occupied by 2 doors, each 7 ft. high, and 3 ft. 
10 in. wide, by 2 windows, each 5 ft. 4 in. high, and 3 ft. 
6 in. wide, by a fireplace 5 ft. square, and by a mop-board 
10 in. wide, extending entirely around the bottom of the 
room, except the spaces occupied by the doors and fire¬ 
place ? Ans. 57.543, about. 

Figure 9. 

8. Find the surface of 
figure 9, divided by the 
dotted lines into 3 triangles, 
whose bases and altitudes 
are marked. 

Ans. 4 A. 3 qrs. 10£ sq. 
rods. 


9. How many yards of carpeting, 1 yd. wide, will be 

necessary for a room 20 ft. long, and 18 ft. 9 in. wide, if 
we deduct the space occupied by the hearth, which is 5 ft. 
long, and 4 ft. 2 in. wide ? Ans. 39.352 nearly. 

10. How much cloth f of a yd. wide, is equal to 26f 

yds. 1^ yd. wide ? * Ans. 40£ yds. 

11. How many square miles, and how many acres are 
there in 6 miles square ? 

Ans. 36 sq. miles, and 23,040 acres. 

12. How much land is there in a piece 100 ft. long, the 

breadths measured every 20 ft. being as follows ; 7 ft., 11 
ft., 5 ft., 6 ft., 16 ft., and 19 ft. ? Ans. 1,020 sq ft 



212 


MENSURATION. 


Lesson 171. 

'Figure 10. 

A square block like figure 10, 
is called a cube. 

By examining the figure, we 
find as we did in Compound Num¬ 
bers, that the surface of the top 
contains 3 times 3, or 9 sq. ft. 
If we take a piece off from the 
top 1 ft. thick, we find the quan¬ 
tity in it to be once 3 multiplied 
by 3, or 9 cubic ft. ; if we take 
a piece off 2 ft. thick, we find the 
quantity to be 2 times 3 multi¬ 
plied by 3, or 18 cubic ft. ; and if we take the whole we 
find the quantity to be 3 times 3 multiplied by 3, or 27 
cubic ft. 

Therefore, to get the quantity in a cube, 

Multiply the length , breadth, and thickness together. 

A body of uniform shape 
and size, whose ends are cut 
off perpendicular to its length, 
is called a right prism , or sim¬ 
ply a prism. Thus, a brick 
is a prism, a square stick of 
timber is a prism, a stick of 
timber hewed 3 square, that 
is, triangular, like figure 11, 
is a prism, also a stick hewed 
5, 6, 7, 8, &c. square, is a 
prism, &c. 

However, if the body be round, like figure 12, it is call¬ 
ed a right cylinder , or simply a cylinder. Thus, a grind¬ 
stone is a cylinder, a round stick of timber is a cylin¬ 
der, &c. 


Figure 11. Figure 12. 




What is called a cubq ? 

What do we find by examining figure 10, lesson 171 ? 

How do we get the quantity in a cube ? 

What is called a right prism, or simply a prism? What things are 
prisms ? 

What is called a right cylinder, or simply a cylinder ? What things 
are cylinders ? 









































MENSURATION. 


213 


Now if we take a piece out of figure 10 in the shape of 
a prism, say 2 ft. wide, 1 ft. thick, and 3 ft. long, it is 
plain that the surface of the end multiplied by 3, gives the 
number of cubic feet the piece contains. A prism of 3 
sides, or of 5, 6, 7, &c. sides, or a cylinder, is evidently 
equal to a prism whose ends contain the same surface, but 
which are square. 

Therefore, to get the quantity in a prism or cylinder, 
Multiply the surface of one end by the length. 


Figure 13. Figure 14. 




When the ends of 
a prism or cylinder 
are not perpendicu¬ 
lar to its length, but 
are parallel to eaclr 
other, as in figures 
13 and 14, the prism 
is called an oblique 
prism, and the cyl¬ 
inder an oblique cyl¬ 
inder. 


If we cut a piece square off from the bottom of an ob¬ 
lique prism, or an oblique cylinder, and place it on the 
top, as the dotted lines in figures 13 and 14 represent, we 
form a right prism, or a right cylinder, of the same length 
and size as the oblique prism, or oblique cylinder. The 
same can be done in an oblique prism whose ends are tri¬ 
angular, or in any other shape. 


Therefore, to get the quantity in an oblique prism, or 
cylinder, 

Multiply the surface of one end of a right prism, or cylin¬ 
der, of the same size, by the length. 


How do we proceed to find a method of obtaining the quantity in a 
prism, or cylinder ? ' 

How do we get the quantity in a prism, or cylinder? 

What is called an oblique prism, or an oblique cylinder ? 

How do we proceed to find a method of obtaining the quantity in an 
oblique prism, or cylinder ? 

How do we get the quantity in an oblique prism, or cylinder? 




MENSURATION. 


214 


Lesson 172. 


Figure 15. 



Figure 16. Figure 17. 


Figure 18. 





A body that tapers regularly from its bottom, or base, 
till it comes to a point, like figuresl5, 16, and 17, is called 
a pyramid. If the base, however, be round, as in figure 
18, the body is called a cone. 

If the top of a pyramid or cone be cut off parallel to the 
base, what remains is called the frustum of a pyramid or 
cone. 

It is plain that all pyramids and cones, which have the 
same height, and whose bases have equal surfaces, are 
equal in size, since they all taper regularly from their 
bases. 


Figure 19. To discover how to measure a pyramid, 

take off part of a 3-sided prism, like fig¬ 
ure 19, by cutting from the corner B 
down to D and F ; cut the figure that re¬ 
mains from B through to A and F. The 
first part we cut off, is a pyramid that has 
the same base as the lower end, or base 
of the prism, and is as high as the prism. 
That part of the remainder, cut off at the 
right, is also a pyramid, which, when 
placed upside down, has the same base as 
the upper end of the prism, and is as high as the prism. 



What is called a pyramid P A cone ? 

What is called the frustum of a pyramid or cone ? 

What is said of all pyramids and cones which have the same height, 
and whose bases have equal surfaces ? 

How do we proceed to find a method of obtaining the quantity in a 
pyramid or cone ? 




















MENSURATION. 


215 


These two pyramids, then, are of the same size, since their 
bases and heights are equal. With regard to the third 
piece, cut off at the left, let us first suppose the prism turn¬ 
ed down on the further side ; the pyramid cut off at the 
right, now has a base composed of half of the further side, 
or face of the prism, and the piece, or pyramid, cut off at 
the left, has a base just as large, being composed of the 
other half; besides, it is just as high, since it tapers regu¬ 
larly up to the point B, the present top ef the pyramid cut 
off at the right ; it is therefore just as large. The one at 
the right being just as large as the first one cut off, it fol¬ 
lows that a triangular prism is composed of three pyra¬ 
mids, equal to the pyramid whose base is one end of the 
prism, and whose height is the same as the length of the 
prism. 

Therefore, as the quantity contained in a prism is equal 
to the product of the surface of one end by the length, 

The quantity contained in a pyramid, or cone, is equal to 

One third of the product of its base by its height. 

The surface of a sphere, globe, or 
ball, like figure 20, has been found 
by calculation to be equal to 

The product of its circumference by 
its diameter. 

This being the case, let us suppose 
a globe divided into a great number 
of pyramids, very small indeed, whose 
bases are at the surface of the globe, 
and tops in the centre, the height of each being the radius, 
or ± of the diameter. The quantity in each pyramid is 
equal to £ of the product of the radius by the base. 

Therefore, the quantity in all of the pyramids, or in the 
whole globe is equal to 

One third of the product of the radius by the surface. 


Figure 20. 



What is the quantity contained in a pyramid or cone equal to ? 

What has the surface of a sphere, globe, or ball been found by calcu¬ 
lation to be equal to ? 

How do we proceed to find a method of obtaining the quantity in a 
globe ? 

To what is the quantity in a globe equal ? 





216 


MENSURATION. 


We can measure a large body of almost any shape by 
dividing it into prisms, pyramids, and frustums, and the 
quantity in the several prisms, pyramids, and frustums, 
will be the quantity contained in the body. 

If the body be very irregular, like a large rock, find the 
average length, breadth, and thickness, and then find the 
quantity in it as though it was a prism. 

A small irregular body can be measured by immersing 
it in a vessel containing water enough to cover it, say in a 
cylindrical vessel, like a pail, tub, &c. ; the space filled by 
the rising of the water will evidently be the quantity con¬ 
tained in the body. 


Lesson 173. 

1. How many cubic feet are there in a pile of bricks in 

the form of a cube, 6 ft. 6 in. long, 6 ft. 6 in. wide, and 6 
ft. 6 in. high ? Ans. 274|. 

2. How many cubic feet, and how many feet of wood are 
there in a cubic pile of wood, measuring 4 ft. on each side ? 

Ans. 64 cubic ft. and 4 ft. of wood. 

3. How many cords of wood are there in a load 8 ft. 
long, 4 ft. 5 in. wide, and 5 ft. 4 in. high ; and what is it 
worth at $6 a cord ? Ans. 1.4575 C. and it is worth $8.74£. 

Explanation. We cafinot in practice get the length, 
width, or height of a load of wood nearer than T V of a foot, 
so in measuring wood when we change inches to decimals, 
we should omit all decimals less than .1 of a foot. 

4. A pile of wood containing 5 C. 5 ft. of wood, is 20 ft. 

long, and 4 ft. wide ; how high is it ? Ans. 9 ft. 

5. What quantity is there in a load of wood 7 ft. 2 in. 
long, 4 ft. 6 in. wide, and 4 ft. 7 in. high ? 

Ans. 1 C. 1.315 ft. 

6. How many cords of wood are there in a pile 30.4 ft. 

long, 4 ft. wide, and 4.3 ft. high ? Ans. 4.085 C. 

7. A stick of timber is 35 ft. 8 in. long, 2 ft. wide, and 1 
ft. 3 in. deep ; how much is it worth at $9 a ton ? 

Ans. $16.05. 

8. A p'lck of timber, hewed 3 square, is 28 ft. long, the 


How can we measure a large body of almost any shape ? 
What if the body be very irregular, like a large rock ? 
How can a small irregular body be measured P 



MENSURATION. 


217 


base of the triangle composing the end is 3 ft., and the alti¬ 
tude is 1 ft. 4 in. ; how many cubic feet does the stick con¬ 
tain ? Ans. 56. 

9. How many cubic feet are there in a barn 40 ft. square 
at the bottom, and measuring 20 ft. from the ground to the 
eaves, and 35 ft. from the ground to the top of the roof ? 

Ans. 44,000. 

10. A laborer dug a cellar 30 ft. long, 22 ft. wide, and 8 

ft. deep, and was paid 6 cents for every cubic yard exca¬ 
vated ; what sum did he receive ? Ans. $11.73. 

11. How many cubic feet are there in a grindstone 5 in. 

thick, and 4 ft. in diameter, making no allowance for the 
eye ? Ans. 5.236, nearly. 

12. How many tons are there in a pine log 42 ft. long, 

and 4 ft. in diameter ? Ans. 10.556, nearly. 


Lesson 174. 

1. There is a wedge 1 ft. 2 in. from the point to the centre 
of the head, the head is 2£ in. thick, and the breadth of 
the wedge is 3 in. ; how many cubic inches are there in it ? 

Ans. 52.5. 

2. The ends of a stick of timber 20 ft. long, 1£ ft. broad, 

and l-£ ft. thick, are sawed off parallel to each other, but 
bevelling, like figure 13, lesson 171 ; how much is the 
stick worth at $8 a ton ? Ans. $6. 

3. How many cubic feet are there in a cylindrical stick 
of timber, 41 ft. long, and 2 ft. 3 in. in diameter, whose 
ends are cut off parallel to each other, but oblique to the 
direction of the stick, like figure 14, lesson 171 ? 

Ans. 163.019, about. 

4. Supposing the largest Egyptian pyramid to be 700 ft. 
square at the base, and 500 ft. high, how many cubic 
yards does it contain, and how much would the cost of 
erecting it be at $2 a cubic yard ? 

Ans. it contains 3,024,691.358 cubic yds., about, and 
the cost of erecting would be $6,049,382.72. 

5. The surface of the base of a triangular pyramid is 

21.25 sq. ft., and the height is 18.33 ft. ; how many cubic 
feet does it contain ? Ans. 129.8375. 

6. A certain church has a steeple shaped like a cone, 
12 ft. in diameter at the base, and 44.5 ft. high ; how 
many cubic feet are there in it ? Ans. 1,677.609, about. 

7. A steeple has a gilded ball on the top 2.25 ft. in di- 

19 


218 


MENSURATION. 


ameter ; how many square feet of gilding are there on the 
ball, and how many cubic feet are there in it ? 

Ans. 15.904 sq. ft. of gilding, about, and 5.964 cubic ft. 
in it, about. 

8. How many cubic feet are there in a rock, the aver¬ 

age length of it being 18.5 ft., breadth 11 ft., and thickness 
9.33 ft.? Ans. 1,898.655. 

9. There is a churn 1.5 ft. in diameter at the bottom, .8 

of a ft. in diameter at the top, and 3.5 ft. high ; how many 
cubic feet of cream will it hold ? Ans. 3.748, nearly. 

Explanation. How much is the diameter of the churn 
contracted in rising 3.5 ft. ? How many feet then must it 
rise to taper out to a point, and form a cone ? How many 
cubic feet will the whole cone contain ? How many cubic 
feet will the little cone formed on top of the churn con¬ 
tain ? Then how many cubic feet does the churn contain ? 

10. How many wine gallons does a churn contain that is 

16 in. in diameter at the bottom, 10 in. in diameter at the 
top, and 36 in. high ? Ans. 21.053, nearly. 

Lesson 175. 

1. There is a tub 3 ft. deep, 4 ft. in diameter at the top, 

and 3 ft. at the bottom ; how many wine gallons does it 
contain ? Ans. 217.38, about 

Explanation. Imagine it to continue downward until it 
comes to a point and forms a cone. 

2. The top of a pyramid was taken off so as to leave a 
portion 18 At. high, 8 ft. square at the base, and 2 ft. 
square at the top ; how much did the portion left contain ? 

Ans. 504 sq. ft. 

3. How many square feet of surface has a cubic block 
of wood, each side of which is 2 ft. square ? 

Ans. 24 sq. ft. 

4. How many square feet of boards, 1 in. thick, will 

make a box measuring on the outside 4 ft. long, 1.5 ft. 
wide, and 2 ft. high ? Ans. 31f. 

Explanation. Observe that the boards at the top, bot¬ 
tom, and sides overlap those at the ends 1 inch, and that 
the boards at the top and bottom overlap those at the sides 
1 inch. 

5. What is the surface of a 3-sided prism, not including 


MENSURATION. 


219 


the ends, which is 12 ft. long, one face or side being 2£ ft 
broad, another 1£ ft., and the third l ft. ? Ans. 60 sq~. ft. 

6. How many square feet of bark are there on a log 20 

ft. long, and 3 ft. in diameter ? Ans. 188.495, about. 

Note. Suppose the convex surface of a cylinder to be divided into 
oblong squares of the same length as the cylinder, and very narrow in¬ 
deed ; if we now multiply the breadth of each by the length, we get its 
surface. Therefore, 

If we multiply the sum of all the breadths, or the entire circumference 
of the cylinder, by the length, xce get the whole convex surface. 

7. There is a pyramid with a base 13 ft. square, and the 
altitude of each triangle composing the sides is 23 ft. ; 
what is the surface of the pyramid, not including the base ? 

Ans 598 sq. ft. 

8. What is the surface of a cone, not including the base, 

the slant height being 18 ft., and the diameter of the base 
6 ft. ? Ans. 169.646 sq. ft., nearly. 

Note. Suppose the convex surface of a cone to be divided into a 
great number of triangles, very small indeed, the tops of which shall be 
at the top of the cone, and the bases at the bottom; it is plain that the 
surface of each of the triangles will be equal to one half of the product 
of its base multiplied by the slant height of the cone. Therefore, 

If we take half the product of the sum of all the bases , or the circumfer¬ 
ence of the base of the cone, multiplied by the slant height of the cone, we 
get the surface of all the triangles, or the whole convex surface of the cone. 

9. A cone 4 ft. in diameter at the base is cut off where 
the diameter is 2 ft. ; what is the convex surface of the re¬ 
maining part, if the slant height of it is 7 ft. ? 

Ans. 65.973 sq. ft., about. 

Explanation. What was the convex surface of the 
whole cone ? Of the part cut off at the top ? 

10. If the earth or globe is 7,920 miles in diameter, how 

many square miles are there in the surface of the northern 
half of it ? Ans. 98,530,315.488 sq. miles. 

Lesson 176. 

Boards are usually sawed 1 inch thick ; and when we 
say 5, 10, 20, &c., feet of boards, we mean 5, 10, 20, &c., 
square feet 1 inch thick. 

How do we proceed to find a way to obtain the convex surface of a 
cylinder ? 

How do we get the convex surface of a cylinder ? 

How do we proceed to find a way to obtain the convex surface of a 
cone ? 

How do we get the convex surface of a cone ? 

How are boards usually sawed ? What do we mean when we say 5, 
10, 20, See., feet of boards? 



220 


MENSURATION. 


Planlcs and joists are sawed thicker ; when we say 
there are 5, 10, 20, &c., feet in a plank or joist, we mean 
there are 5, 10, 20, &,c., feet of boards 1 inch thick in the 
plank or joist. 

Round timber is often measured by the following arbi¬ 
trary rule. 

Multiply £ of the average girt of the log by itself \ and this 
product by the length. 

Note. This rule is incorrect, giving about § less than the true quanti¬ 
ty ; l being allowed for waste from knots, crooks, &c. 

1. How many feet are there in a pile of 40 boards, each 
being 19 ft. 4 in. long, 1 ft. 3 in. wide, and 1 in. thick ? 

Ans. 966|. 

2. How many feet are there in 2 planks 33 ft. long, 2 ft. 

3 in. wide, and 3 in. thick ? Ans. 445.5. 

3. A man bought 20 pine joists, each of which was 18 ft. 

long, 5 in. wide, and 3 in. thick, at $20 a thousand feet ; 
what did they cost him ? Ans. $9. 

4. What sum must I give for 50 pine planks 30 ft. long, 
2.5 ft. wide, and 4 in. thick, at $16 a thousand feet ? 

Ans. $240. 

5. There is a pine log 30 ft. long, the average girt of 

which is 10 ft.; how many cubic feet does it contain by the 
preceding arbitrary rule ? Ans. 187.5. 

6. If you work by the preceding arbitrary rule, how- 

many tons will there be in a stick of round timber 40 ft. 
long, with an average girt of 8 ft., 40 cubic ft. being reck¬ 
oned to a ton ? Ans. 4. 

7. What sum must you receive for building a wall 87 ft. 

long, 7 ft. thick, and 12.5 ft. high, at $1.80 a perch, there^ 
being 24J- cubic feet in a perch ? Ans. $553.64. 

8. There is a brick house 48 ft. long, and 26 ft. wide ; 
the walls are 19 ft. high, and 1 ft. thick ; the gable ends 
are 15 ft. high, and 8 in. thick ; there are 2 doorways 8 ft. 
high, and 5 ft. wide, and 18 windows in the lower part, and 

4 in the gable ends, each being 6 ft. high, 3£ ft. wide ; 

now how many bricks are there in the house, making no 
allowance for the lime, a brick being 8 in. long, 4 in. wide, 
and 2 in. thick ? Ans. 67,014. 

9. How many cubic inches are there in a marble image. 

How are planks and joists sawed ? What do we mean when we sag 
there are 5, 10, 20, &c., feet in a plank or joist ? 

Recite the arbitrary rule by which round timber is usually measured. 

What is said of the correctness of this rule ? 



GAUGING. 


221 


which, being immersed in the water contained in a pail 10 
in. in diameter, caused the water to rise 1 in. ? 

Ans. 78.54, nearly. 

10. A stone being immersed in some water contained in 
a box 12 in. wide and 20 in. long, caused the water to rise 
2£ in. ; how many cubic inches were there in the stone ? 

Ans. 600. 


GAUGING. 

Lesson 177. 

To gauge a cask, that is, to find how much it holds, 

Find on the inside in inches , the length , the hung diameter , 
and the head diameter ; then if the staves he much curved be¬ 
tween the hung and head , add to the head diameter § of the 
difference between the hung and head diameters; hut if the 
staves be but little curved , add of the difference ; the sum is 
found by experience, to he about the diameter of a cylinder of 
the same length and capacity as the cask; the quantity con¬ 
tained in the cylinder is the quantity the cask will hold. 

Note 1. We are generally obliged to take all the dimensions, except 
the bung diameter, on the outside; when such is the case, we must 
make a proper allowance, say from 1 to 2 inches for the thickness of the 
two heads ; and the head diameter, measured within the chimes, must 
be diminished from .3 to .6 of an inch, on account of the greater thick¬ 
ness of the stave inside of the head. Observe that the average of the 
two head diartieters must be taken when they are unequal. 

Note 2. In gauging, we can consider the circumference of a circle 
3.14 times the diameter; this will be near enough. 

1. How many wine gallons will a cask hold, whose 
staves are moderately curved, the head diameter being 16 
in., bung diameter 20 in., and length 28 in. ? - 

Ans. 32.2, about. 

Explanation. What is the diameter of a cylinder of the 
same length and capacity as the cask ? How many cubic 
inches does such a cylinder hold ? How many wine gal¬ 
lons are there in this number of cubic inches ? 


What is the rule for gauging a cask ? 

How are we generally obliged to take all the dimensions of a cask P 
When such is the case, what allowances must be made ? If the two 
head diameters are unequal, what is to be done ? 

In gauging, how can we consider the circumference of a circle ? 

19* 




222 


GAUGING. 


2. How many bushels of salt will a cask hold, whose 
length is 42 in., head diameter 26 in., and bung diameter 
33 in., the staves being much curved ? Ans. 14.4, about. 

3. How many imperial gallons of milk are there in a 
cask, whose length is 25 in., bung diameter 19 in., and 
head diameter 16 in., the staves curving but little ? 

Ans. 22.4, about. 

4. What number of beer gallons will a cask hold, whose 
length is 33 in., bung diameter 22 in., and head diameter 
16 in., the staves curving moderately ? Ans. 35.3, nearly. 

5. How many bushels of wheat are there in 500 casks, 
the length of each being 48 in., the bung diamefer 36 in., 
the head diameter 28 in., and the staves much curved ? 

Ans. 9,734.6, about. 

Lesson 178. 

To find the quantity of liquor in a cask which is not full. 

Set the cask upright ; then if the height of the liquor he not 
more than % of the height of the cask , consider the space filled 
as forming the frustum of a cone, and calculate the quantity 
accordingly, if the height of the liquor be more than but 
not more than £ of the height of the cask, consider the space 
filled as \ of a cask, and calculate the quantity accordingly. 

Note 1. When the liquor fills more than k of the cask, measure the 
hollow space by the preceding rule, and then to find the amount of 
liquor, subtract this quantity from the whole contents, of the cask. 

Note 2. When casks lie on one side, guagers employ certain arbitrary 
rules to find the empty space in them. These rules, which are not very 
correct, may be learned by practice. 

1. There is a barrel 34 in. long which contains a-quan¬ 
tity of molasses; when placed on one head, the surface of 
the liquor is 8 in. high ; the head diameter of the barrel is 
16 in., and the diameter at the surface of the molasses is 
18 in. ; what quantity of molasses does the barrel contain ? 

Ans. 7.9 gals., nearly. 

2. How much wine is there in a cask standing upright, 
the height of the cask being 30 in., the height of the wine 
14 in., the head diameter 15 in., and the diameter at the 


How do we find the quantity of liquor in a cask which is not full ? 
What is done when the liquor fills more than £ of the cask ? 

How is the empty space found when casks lie on one side P 



TONNAGE OF VESSELS. 


223 


surface of the liquor 18 in., the staves being much curv¬ 
ed ? Ans. 13.7 gals., about. 

3. A cask of water 29 in. long, when placed on one 
head, lacks 6 in. of being full ; the head diameter is 15 in., 
the bung diameter 18 in., and the diameter at the surface 
of the water 16 in. ; how many imperial gallons of water 
are there in it, if the staves are much curved ? 

Ans. 19.6, about. 

4. How many bushels of wheat are there in a hogshead 

standing upright, with staves moderately curved, the height 
being 46 in., the height of the wheat 25 in., the head di¬ 
ameter 26 in., the bung diameter 36 in., and the diameter 
at the surface of the wheat 34 in. ? Ans. 9.9, about. 


TONNAGE OF VESSELS. 

Lesson 179. 

The following arbitrary rule is employed, at present, to 
find the number of tons a vessel will carry. 

Subtract of the breadth of the vessel from the length ; 
then multiply the remainder , the breadth , and the depth together , 
and divide the product by 95. 

The length,, breadth, and depth are measured in feet 
and decimals of a foot, as follows. The length from the 
fore part of the main stem to the after part of the stern 
post, above the upper deck ; the breadth at the broadest 
part above the main wales ; the depth in single decked ves¬ 
sels, from the under side of the deck plank to the ceiling 
of the hold ; but in double decked vessels £ of the breadth 
is called the depth. 

Note. In some places ship carpenters, to find the number of tons, 
multiply the length of the keel, the breadth, and depth together, and 
divide the product by 95. 

1. A ship carpenter built a schooner, which is a single 
decked vessel, on contract ; the length was 84.4 ft., the 
breadth 24 ft., and the depth 10 ft. ; how many tons were 
to be paid for ? Ans. 176f^. 

Recite the arbitrary rule employed, at present, to find the number of 
tons a vessel will carry. 

How are the length, breadth, and depth, measured ? 

In some places how do ship carpenters find the number of tons? 




224 


SQUARE ROOT. 


2. What is the tonnage of a double decked vessel, the 
length being 106.2 ft., and the breadth 27 ft. ? 

Ans. 345f§ tons. 

3. What is the tonnage of a double decked vessel, the 
length being 150 ft., and the breadth 36 ft. ? 

Ans. 875£f tons, about. 

4. How many tons will a single decked vessel carry, 
whose length is 75 ft., breadth 20 ft., and depth 9 ft. ? 

Ans. I19f|. 

5. What number of tons are there in a sloop, which is a 

single decked vessel, 60 ft. long, 18.5 ft. wide, and 7.2 ft. 
deep ? Ans. 68ff, about. 

6. How many tons are there in a double decked vessel, 
the length being 190 ft., and the breadth 44 ft. I 

Ans. 1,667, nearly. 


SQUARE ROOT. 

6 

Lesson 180 . 

If a number be multiplied by itself, the product is called 
the square of that number ; thus, 4 is the square of 2, 25 
is the square of 5, &c. 

That number, which multiplied by itself will produce a 
certain other number, is called the square root of this other 
number ; thus, 2 is the square root of 4, 5 the square root 
of 25, &c. 

The square of a number is easily found, being obtained 
by multiplying the number by itself, but it is more difficult 
to get the square root of a number ; however, when the 
square root is a whole number, not exceeding 10, it is 
readily found by trying a few times. 

Examples to he performed in the mind. 

What is the square root of 49 ? Of 36 ? Of 64 ? Of 25 ? 
Of 81 ? Of 16 ? Of 100 ? Of 9 ? Of 4 ? 


What is called the square of a number ? What is the square of 2 ? 
Of 5? 

What is called the square root of a number P What is the square 
root of 4 ? Of 25 ? 

What is said of the ease of finding the square and square root of a 
number ? 




SQUARE ROOT. 


225 


When the square root of a number exceeds 10, this man¬ 
ner of obtaining it is tedious ; for instance, if the square 
root of 169, of 256, of 361, or of 625, is required, we are 
obliged to try many times before we find a number which, 
multiplied by itself, will produce either of them. In order 
to discover a method of getting the square root with facil¬ 
ity when it -exceeds 10, we first observe how we find the 
square of a number more than 10, say of 27. To make 
the operation more plain, we separate 27 into two parts, 
20 and 7. 


OPERATION. 


20 

20 


400 


140 

140 


49 


400 280 49 


Adding the several products. 
400 
280 
49 


729 square of 27. 


Explanation. We see that 
the square of 27 consists of 
20 times 20, of 20 times 7, 
of 7 times 20, and of 7 
times 7. 

Therefore, the square of 
a number containing two 
figures, consists of 

The square of the tens , 
twice the product of the tens 
hy the units , and the square 
of the units. 

Now the square of 10 is 
100, the square of 100 is 
10,000, the square of 1,000 
is 1,000,000, &lc. It appears 


then, that the square of units is found in the two right hand 
figures, because the square of 10, or 100, is the smallest 
number possible consisting of three figures ; the square of 
tens is found in the two next figures, because the square 
of 100, or 10,000, is the smallest number possible consisting 
of five figures. It can be shown in the same way, that the 
square of hundreds is found in the two figures at the left of 
the square of the tens ; that the square of thousands is 
found in the two figures at the left of the square cf hun¬ 
dreds, &c. 


What if the square root of a number exceeds 10 ? 

How do we proceed to discover a method of getting the square root 
with facility when it exceeds 10 ? 

What does the square of 27 consist of? 

What then does the square of a number containing two figures con¬ 
sist of? 

Where is the square of units found ? Why ? Where is the square 
of tens found ? Why ? What else can be shown ? 






226 


SQUARE ROOT. 


Let us find the square root of 729. 

Explanation. The square of the units 
of the root is in 29, and the square of 
the tens in 7. The greatest square in 7 
is 4, the square root of which is 2 ; this 
root we place at the right like a quo¬ 
tient, subtract 4, the square of 2, from 
7, and bring dowrr the 29. The remain¬ 
ing number, 329, contains twice the product of the tens by 
the units, and the square of the units; now twice any num¬ 
ber of tens multiplied by units, gives nothing less than 
tens , so if we divide the 32 tens in 329 by twice the tens 
already obtained, or 4, we shall get the number of units, 
or too large a number, since the square of the units usual¬ 
ly increases the tens a little. In fact, we get 8, which we 
place at the right of the 2, and also at the right of the 4 
we divided by ; we now multiply 48 by 8, which if 8 be 
right gives the square of the units and the product of twice 
the tens by the units. The result being 384, greater than 
329, we conclude that 8 is too large, we rub it out, substi¬ 
tute 7, and multiplying 47 by 7, obtain 329; 7 then is right, 
and 27 is the square root of 729. 

Lesson 181. 


OPERATION. 

729(27 Ans. 
4 


47)329 

329 


What is the square root of 1,447,209 ? 


OPERATION. 

1447209(1,203 Ans. 
1 


22 ) 


44 

44 


Explanation. The square of 
the units of the root is in the two 
right hand figures ; the square 
of the tens is in the two next 
figures, and so on. We there¬ 
fore begin at the right, and divide 
1,447,209 by dots into parts, of 
two figures each. We find the 
root of the two left hand parts, or 
of 144, just as before. This is 
evidently correct, since the square of the two first figures 
in the root, is contained in 144. To get the remaining 
part of the root, consider the 12 already obtained as the 
tens, bring down 72, and divide 7 by twice 12, or 24, 


2403 ) 7209 
' 7209 


Explain how you find the square root of 729. 
Explain how you find the square root of 1,447,209. 







SQUARE ROOT. 


227 


which is contained in 7, 0 times ; we place 0 then at the 
right of 12, and considering 120 as the tens, finish as 
before. 

From what precedes, we derive the following rule for 
obtaining the square root; 

Begin at the right, and separate the number, by dots, into 
parts of two figures each. Find the greatest square in the 
left hand part, write its root as you do a quotient, subtract the 
square from the left hand part, and bring down the two next 
figures at the right of the remainder for a dividend. Double 
the root already found for a divisor, and omitting the right 
hand figure of the dividend, find how many times the divisor 
is contained in the rest of it, and place the result at the right 
of the root already found, and also at the right of the divisor. 
Multiply the divisor thus increased, by the last figure in the 
root, subtract the product from the whole dividend, and bring 
down the two next figures, and so on ; but if the product be 
greater than the dividend, diminish the last figure in the root 
until the product shall be equal or less, and then subtract it 
from the dividend. 

When a divisor is not contained in a dividend, the right 
hand figure of which is omitted, write 0 in the root, and at 
the right hand of the divisor, and bring down the next two 
figures, and divide as before. 

To prove the Square Root, 

Multiply the root by itself and the original number will 
evidently be produced if the work be right. 

Lesson 182 . 

1. What is the square root of 256 ? Ans. 16. 

2. What is the square root of 324 ? Ans. 18. 

3. What is the square root of 97,476,129 ? Ans. 9,873. 

4. What is the square root of 1,002,001 ? Ans. 1,001. 

5. What is the square root of 998,001 ? Ans. 999. 

6. A man bought a square house lot, containing 2,025 

sq. ft. ; how long was each side of it ? Ans. 45 ft. 


Recite the rule for obtaining the square root. 

What if a divisor is not contained in a dividend, the right hand figure 
of which is omitted ? 

To prove the square root how do we proceed ? 




223 


SQUARE ROOT. 


7. A farmer determined to set out 784 apple trees in an 

exact square ; how many rows will he have, and how many 
trees in a row ? Ans. 28 rows, and 28 trees in a row. 

8. I have 2 pieces of land, each in the shape of an ob- 

l6ng square ; one is 80 ft. by 37, and the other 32 ft. by 
20 ; how long must the side of a square be to contain as 
much as the two pieces ? Ans. 60 ft. 

9. A general formed 4,624 men into an exact square ; 
how many were there on one side of the square ? Ans. 68. 

10. I wish to put 1,250 sq. ft. of land into an oblong 

square, the length of which shall be twice the breadth ; 
what will the breadth be ? Ans. 25 ft. 

Lesson 183. 

1. What is the square root of 0.36 ? 

Explanation. In the first 
place we observe that a num¬ 
ber must contain twice as 
many decimals as its root ; 
for the square of the root, 
that is, the root multiplied by 
itself, contains twice as many 
decimals as the root. See 
Decimal Fractions, lesson 
80. The number must there¬ 
fore contain two, four, six, 
eight,or ten,&c.,decimals, and 
it contain four, by annexing a 
0, which does not alter its value. See Decimal Fractions, 
lesson 75. It is evident we must now proceed to find the 
root as in whole numbers. After obtaining two figures of 
the root, we have a remainder 36, and annexing two 0s for 
a dividend, which amounts to the same thing as bringing 
them down, had they been originally placed at the right 
of .0360, we obtain another figure. We can get as 
many more in this way as we think fit. There are evi¬ 
dently three decimals in the root, which we have obtained, 
as there are six in the number, including those annexed to 
the remainder, 36. 


OPERATION. 

,03*60(.189 about. Ans. 
1 

28 ) 260 
224 


369 ) 3600 
3321 


279 remainder. 

* .036 having three, we make 


Explain how example 1, lesson 183, is performed. 





SQUARE ROOT. 


229 

Therefore, when a number contains decimals, if they do 
not consist of two, four, six, or eight, &c., figures, 

Annex a 0, and observe that there will be half as many 
decimals in the root , as in the number including the Os annex¬ 
ed to the remainders. 


2. What is the square root of 3.3 ? 


OPERATION*, 

3.30(1.816, about. Ans. 
1 

28) 230 
224 


361)600 

361 


Explanation. There be¬ 
ing but one decimal, we 
make another, and then get 
the root as in whole num¬ 
bers, annexing two Os to 
each remainder, and con¬ 
tinuing the operation until 
the root is sufficiently accu¬ 
rate. 


_ A separating dot comes 

3626) 23900 between decimals and whole 

21756 numbers, as there always 

_ will, since the decimals are 

2144 remainder. made to consist of two, four, 

six, or eight, &c., figures. 
Moreover, it is plain that as soon as decimals are brought 
down, we get decimals in the root, for the square of whole 
numbers never produces decimals, any more than the 
square of decimals produces whole numbers. 

Many whole numbers do not have an exact square root 
in whole numbers ; as 2, 3, 5, 6, 7, 8, &c. 


What is done when a number contains decimals, if they do not con¬ 
sist of two, four, six, or eight, &c., figures? 

How many decimals will there be in the root ? 

Explain how example 2, lesson 183, is performed. 

Where will a separating dot always come ? Why ? When do we get 
decimals in the root ? Why? 

What is said of whole numbers having an exact root? 


20 






230 


SQUARE ROOT. 


3. What is the square root of 2 ? 

operation. Explanation 


2(1.414, about. Ans. 
1 


24)100 

96 


281) 400 
281 


After finding 
the greatest root in 2, there is 
1 remainder ; we can evidently 
carry the operation as far as 
proper by annexing two 0s as 
decimals, to each remainder, 
and proceeding just as we should 
were they originally placed at 
the right of 2. 

Although we can obtain a 
root as near as desirable by an¬ 
nexing two 0s as decimals, to 
each remainder, and continuing 
the operation, still if the exact 
root is not found without adopt¬ 
ing this course, it cannot be found by it ; for the last 
figure in each dividend will always be 0, from which the 
units in the square of the next figure in the root are to be 
subtracted, and as the square of no figure can produce a 
number with 0 in the units’ place, there will always be a 
remainder. 


2824)11900 

11296 


604 remainder. 


To multiply a common fraction by itself, that is, to get 
the square of it, we square the numerator and denomina¬ 
tor. See Common Fractions, lesson 71. 


Therefore, to find the square root of a common fraction, 

Find the square root of the numerator and denominator ; 
thus, the square root of T 9 ¥ is £, of f is f, &.c. 

However, in many common fractions, like §, £, &.C., 
we cannot get the root of both the numerator and denomi¬ 
nator in whole numbers. In such cases, and in mixed 
numbers, it is best to change the fractions to decimals 
first, and then get the root. 


Explain how example 3, lesson 183, is performed. 

What is said of obtaining an exact root by annexing two Os as deci¬ 
mals to each remainder, &c. ? 

How do we get the square of a common fraction ? 

How do we find the square root of a common fraction ? What is the 
square root of ^ ? Of | ? 

What course do we take with mixed numbers; and what with com¬ 
mon fractions, when we cannot get*the root of both numerator and de¬ 
nominator in whole numbers ? 






SQUARE ROOT. 

Lesson 184. 


231 


1. What is the square root of .0081 ? Ans. .09. 

2. What is the square root of 628.195 ? 

Ans. 25.0638, about. 

3. What is the square root of 895,372 ? 

Ans. 946.241, nearly. 

4. What is the square root of ? Ans. or |. 

5. What is the square root of ? Ans. .433, about. 

6. What is the square root of 98^ ? Ans. 9.912, about. 

7. A square garden contains 6.25 sq. rods ; what is the 

length of one side of it ? Ans. 2.5 rods. 

8. A field in the shape of an oblong square, 5 times as 
long as it is wide, contains 2,672.05 sq. ft. ; how many 
feet wide is it, and what is its length ? 

Ans. width 23.117 ft., about ; length 115.585 ft., about. 

9. A field containing 61,322 sq. ft. is | as wide as it is 
long ; how wide is it, and what is its length ? 

Ans. width 214.456 ft., about ; length 285.941 ft., about. 
Explanation. It is f as long as it is wide. 

10. What is the square root of .00048 ? 

Ans. .0219, about. 

11. There is a field 20.25 rods square, which I have 
agreed to exchange for another field equally large, but 
which shall be three times as long as it is wide ; what will 
be the length and breadth of the field ? 

Ans. 35.073 ft. length, about ; 11.691 ft. breadth, about 

Lesson 185. 

If a triangle has a square corner, or right 
angle, the square of the side opposite the right 
angle will be equal to the sum of the squares 
of the two sides adjacent the right angle. 

For, if the sides adjacent the right angle 
are equal, as in figure 21, we see that the 
square of one of them contains two trian¬ 
gles, and is half as large as the square of 
the side opposite the right angle, which 
contains four triangles of equal size. If the two sides ad- 


If a triangle has a square corner or right angle, how is the square of 
the side opposite the right angle to the sura of the squares of the two 
sides adjacent the right angle ? Explain it. 


Figure 21. 






232 


SQUARE ROOT. 


jacent the right angle are not equal, it can also be shown 
that the sum of their squares is equal to the square of the 
side opposite the right angle. 

1. A room is 18ft. long, and 15 ft. wide ; what is the 
distance between the opposite corners ? 

Ans. 23.431 ft., nearly. 

2. The wall of a fort is 24 ft. high, and there is a ditch 
beside it 18 ft. wide ; how long must a ladder be to reach 
from the outside of the ditch to the top of the wall ? 

Ans. 30 ft. 

3. If the ladder be 20 ft. long, and the wall 16 ft. high, 

how wide is the ditch ? Ans. 12 ft. 

4. If the ladder be 25 ft. long, and the ditch 15 ft. wide, 

how high will the wall be ? Ans. 20 ft. 

5. A certain field, in the shape of an oblong square, is 
40 rods long, and 36 rods wide ; what is the distance be¬ 
tween the opposite corners ? Ans. 53.814 rods, about. 

6. If you make a square with one side 6 ft. long, and 

the other 8 ft. long, what will be the length of a pole that 
will measure the distance from the end of one side to the 
end of the other ? Ans. 10 ft 

Note. Carpenters usually make a square by fastening two pieces of 
wood together, one 6 ft. long, and the other 8 ft. long, and making the 
distance between the two ends 10 ft. 

7. A carpenter building a house 24 ft. wide, wishes to 

have the gable end 12 feet high; how long must the rafters 
be ? Ans. 16.97 ft., about. 

Explanation. Make a figure of it on your slate. 

8. What is the distance between the opposite corners 
of a square field, containing 2 A. 1 qr. 32 sq. rods ? 

Ans. 28 rods. 

9. I wish to hew the largest square stick possible out 

of a log 16 in. in diameter ; what size will the end of the 
square stick be ? Ans. 11.3 in. square, about. 

Explanation. The diameter of one end of the log is the 
distance between the opposite corners of the square. 

10. If you have one pole 20 ft. long, and another 12 
ft., how long must a third be, so that they may form a 
right angle when put together in the form of a triangle, 
the 20 ft. pole being opposite the right angle ? 

Ans. 16 ft. 


CUBE ROOT. 


233 


CUBE ROOT. 

Lesson 186. 

If we multiply a number and its square together, the 
product is called the cube of that number ; thus, 8 is the 
cube of 2, 27 is the cube of 3, &c. 

That number, which, multiplied by its square, will pro 
duce a certain other number, is called the cube root of this 
other number ; thus, 2 is the cube root of 8, 3 the cube 
root of 27, &c. 

The cube of a number is easily found, being obtained 
by multiplying the number and its square together, but it 
is more difficult to get the cube root of a number ; how¬ 
ever, where the cube root is a whole number, not exceed 
ing 10, it is readily found by trying a few times. 

Examples to be performed in the mind. 

What is the cube root of 343 ? Of 216 ? Of 512 ? Of 
125 ? Of 729 ? Of 64 ? Of 1,000 ? Of 27 ? Of 8 ? 

When the cube root of a number exceeds 10, this man¬ 
ner of obtaining it is tedious ; for instance, if the cube root 
of 2,197, or of 4,096, is required, we are obliged to try 
many times before we find a pumber, which, multiplied by 
its square, will produce either of them. In order to dis¬ 
cover a method of getting the cube root with facility when 
it exceeds 10, we first observe how we find the cube of a 
number more than 10, say of 27. To make the operation 
more plain, we multiply the square of 27 in three parts, as 
we found it in the Square Root, lesson 180, by 27 in two 
parts, 20 and 7. 


What is called the cube of a number ? What is the cube of 2 ? Of 3 ? 

What is called the cube root of a number? What is the cube root 
of 8? Of 27? 

What is said of the ease of finding the cube and cube root of a num¬ 
ber ? 

What if the cube root of a number exceeds 10 ? How do we pro¬ 
ceed to discover a method of getting the cube root with facility, when 
it exceeds 10 ? 


20* 



234 


CUBE ROOT. 


OPERATION. 

400 280 49 Explanation. 8,000 is the 

20 7 cube of 20, 8,400 is formed of 

- the square of 20 multiplied by 

2800 1960 343 7, and of the product of 2 times 

8000 5600 980 20 by 7 multiplied by 20, mak- 

- ing 3 times the square of 20 

8000 8400 2940 343 multiplied by 7 ; 2,940 is form¬ 

ed of the product of 2 times 20 
Adding up the several products, by 7 multiplied by 7, and the 
8 0 0 0 square of 7 multiplied by 20, 

8 4 0 0 making 3 times 20 multiplied 

2 9 4 0 by the square of 7 ; 343 is the 

3 4 3 / cube of 7. 


1 9,6 8 3 cube of 27. 

Therefore, the cube o? a number containing two figures, 
consists of, 

The cube of the tens, three times the square of the tens 
multiplied by the units, three times the tens multiplied by the 
square of the units, and the cube of the units. 

Now the cube of 10 is 1,000, the cube of 100 is 1,000,000, 
the cube of 1,000 is 1,000,000,000, &c. It appears, then, 
that the cube of units is found in the three right hand fig¬ 
ures, because the cube of 10, or 1,000, is the smallest 
number possible consisting of four figures ; the cube of 
tens is found in the three next figures, because the cube of 
100, or 1,000,000, is the smallest number possible consist¬ 
ing of seven figures. It can be shown in the same way, that 
the cube of hundreds is found in the three figures at the 
left of the cube of tens ; that the cube of thousands is 
found in the three figures at the left of the cube of hun¬ 
dreds, &c. 

Let us find the cube root of 19,683. 


Explain what the cube of 27 consists of? 

What then does the cube of a number, containing two figures, con¬ 
sist of? 

Where is the cube of units found? Why? Where is the cube 
of tens found ? Why ? What else can be shown ? 






CUBE ROOT. 


235 


OPERATION. 


19683 ( 27 Ans. 
8 


20 20 tens. 

20 3 


7 

7 


1200 ) 11683 
11683 


400 

3 


49 square of 


60 


49 

7 


the units. 


3 times the square of the tens 1200 540 

7 240 


343 


8400 2940 

2940 
343 


11683 


Explanation. The cube of the units is in 683, and the 
cube of the tens in 19. The greatest cube in 19 is 8, the 
cube root of which is 2 ; this root we place at the right, 
like a quotient ; subtract 8, the cube of 2, from 19, and 
bring down the 683. The remaining number, 11,683, con¬ 
tains 3 times the square ofthe tens multiplied by the units, 
3 times the tens multiplied by the square of the units, and 
the cube of the units ; now 3 times the square of any num¬ 
ber of tens multiplied by units, gives nothing less than 
hundreds; so if we divide the 116 hundreds in 11,683 by 3 
times the square of the 2 tens, or 1,200, we shall get the 
number of units, or too large a number, since 3 times the 
tens multiplied by the square of the units, and the cube of 
the units, usually increase the hundreds considerably. In 
fact, we get 9, which, on the proper trial, is found too 
large, as well as 8. Let us try 7 ; placing it at the right 
of 2, to find whether it is right or not, we add together 3 
times the square of the tens multiplied by 7, 3 times the 
tens multiplied by the square of 7, or 49, and the cube of 
7, or 343 ; the sum making 11,683, the same as the re¬ 
maining number, we conclude that 7 is right, and that 27 
is the cube root of 19,683. 


Explain how you find the cube root of 19,683. 







236 


CUBE ROOT. 


Lesson 187 

What is the cube root of 1,740,992,427 ? 

OPERATION. 

1740992427 (1,203 Ans. 

1 

300 ) 740 
728 


4320000 ) 12992427 
12992427 


Explanation. The cube of the units of the root, is in the 
three right hand figures ; the cube of the tens is in the 
three next figures, and so on. We therefore begin at the 
right, and divide 1,740,992,427, by dots, into parts of three 
figures each. We find the root of the two left hand parts, 
or of 1,740, just as before ; this is evidently correct, since 
the cube of the two first figures in the root is contained in 
1,740. To get the remaining part of the root, consider 
the 12 already obtained as the tens, bring down 992, and 
divide 12,992 by 3 times the square of 12 tens, or 43,200, 
which is contained in 12,992, 0 times ; we place 0 then at 
the right of 12, and considering 120 as the tens, finish as 
before. 

From what precedes, we derive the following rule for 
obtaining the cube root ; 

Begin at the right, and separate the number, by dots, into 
parts of three figures each. Find the greatest cube in the left 
hand part, write its root as you do a quotient, subtract the cube 
from the left hand part, and, bring down the next three figures 
at the right of the remainder, for a dividend. Each of the re¬ 
maining operations to find the root, is as follows ; Multiply 
the square of 10 times the root already found bp 3, for a divi¬ 
sor, find how many times it is contained in the dividend, and 
place the result at the right of the root already found. Add 
together 3 times the square of the tens, hundreds, fyc. in the 
root, multiplied by the units, 3 times the tens, hundreds, fyc. t 


Explain how you find the cube root of 1,740,992,427. 





CUBE ROOT. 


237 


multiplied by the square of the units, and the cube of the units; 
if the result be greater than the dividend, diminish the last 
figure in the root, until the residt shall be equal, or less, and 
then subtract it from, the dividend. Bring down the next three 
figures at the right of the remainder for a dividend, and pro¬ 
ceed to get the next figure in the root as before, and so on. 

When a divisor is not contained in a dividend, write 0 in 
the root, and bring down the next three figures, and divide 
as before. 

To prove the Cube Root, 

Multiply the root and the square of the root together, and 
the original number will evidently be produced if the work be 
right. 


Lesson 188 . 

1. What is the cube root of 2,197 ? Ans. 13. 

2. What is the cube root of 15,625 ? Ans. 25. 

3. What is the cube root of 9,261 ? Ans. 21 

4. What is the cube root of 2,924,207 ? Ans. 143 

5. What is the cube root of 729,000 ? Ans. 90 

6. What is the cube root of 164,359,469,195,433 ? 

Ans. 54,777. 

7. What is the length of one side of a cubic block of 

wood containing 1,728 cubic inches ? Ans. 12 in. 

8. The quantity of water that passes over the dam 

across a certain stream, is 60 cubic ft. a second ; what is 
the length of a cubic cistern that will contain what passes 
over iti one hour ? Ans. 60 ft. 

9. If you have 13,824 cubic in. of lead, how large a cube 
will it make if melted together ? Ans. a cube 24 in. square. 

10. A lumber dealer has 110,592 cubic ft. of hewed tim¬ 

ber ; what will be the length of a cubic pile to contain the 
whole ? Ans. 48 ft. 


Recite the rule for obtaining the cube root. 

What if a divisor is not contained in a dividend ? 
To prove the cube root how do we proceed ? 



CUBE ROOT. 


. 

Lesson 189. 

1. What is the cube root of .0365 ? 

OPERATION. 

.036500(.331 about. Ans. 
27 


2700 ) 9500 
8937 


326700 ) 563000 
327691 


235309 remainder. 

Explanation. In the first place we observe that a num 
ber must contain three times as many decimals as its root, 
for the cube of the root contains as many decimals as the 
root and its square. The number must therefore contain 
three, six, nine, or twelve, &c. decimals, and .0365 having 
four, we make it contain six, by annexing two 0s which 
does not alter its value. See Decimal Fractions, lesson 
75. It is evident we must now proceed to find the root as 
in whole numbers. After obtaining two figures of the root, 
we have a remainder 563, and annexing three 0s for a divi¬ 
dend, which amounts to the same thing as bringing them 
down had they been originally placed at the right of 
.036500, we obtain another figure. We can get as many 
more in this way as we think fit. There are evidently 
three decimals in the root which we have obtained, as there 
are nine in the number, including those annexed to the re¬ 
mainder, 563. 

Therefore, when a number contains decimals, if they do 
not consist of three, six, nine, &c. figures, 

Annex as many 0s as will make three, six, or nine , fyc. de 
cimals, and observe that there will be one third as many deci¬ 
mals in the root as in the number, including the 0s annexed to 
the remainders. 


Explain how example 1, lesson 189, is performed. 

What is done when a number contains decimals, if they do not con 
sist of three, six, or nine, &c. figures ? 

How many decimals will there be in the root? 







CUBE ROOT. 


239 


2. What is the cube root of 2.41 ? 

OPERATION 

2.410 ( 1.34 about. Ans. Explanation. The num- 
1 - her of decimals being two, 

— we. make it three, and then 

300)1410 get the root as in whole 

1197 numbers; annexing three 

- Os to each remainder, and 

50700)213000 continuing the operation un- 

209104 til the root is sufficiently 

- accurate. A separating dot 

3896 remainder. comes between decimals and 
whole numbers, as there always will, since the decimals 
are made to consist of three, six, or nine, &c. figures. 
Moreover, it is plain that as soon as decimals are brought 
down, we get decimals in the root, for the cube of whole 
numbers never produces decimals, any more than the cube 
of decimals produces whole numbers. 

Many whole numbers do not have an exact cube root in 
whole numbers, as 2, 3, 4, 5, 6, 7, 9, &c. 

3. What is the cube root of 5 ? 


operation. Explanation. After finding 

5( 1.71 nearly. Ans. the greatest root in 5, there is 
1 4 remainder ; we can evidently 

carry the operation as far as 
proper, by annexing three 0s, 
as decimals, to each remain¬ 
der, and proceeding just as 
we should were they originally 
placed at the right of 5. 


300)4000 

3913 


86700)87000 

87211 


Although we can obtain a root as near as desirable by 
annexing three 0s, as decimals, to each remainder, and 
continuing the operation, still, if the exact root is not found 
without adopting this course, it cannot be found by it, for 
the last figure in each dividend will always be 0, from 


Explain how example 2, lesson 189, is performed. Where will a 
separating dot always come? Why? When do we get decimals in 
the root? Why? 

What is said of whole numbers having an exact cube root? 

Explain how example 3, lesson 189, is performed. 

What is said of obtaining an exact root by placing three Os, as deci¬ 
mals, after each remainder, &c. ? 







CUBE ROOT. 


240 

which the units in the cube of the next figure in the root 
are to be subtracted, and as the cube of no figure can pro¬ 
duce a number with 0 in the units’ place, there will always 
be a remainder. 

To get the cube of a common fraction, we cube the nume¬ 
rator and denominator. See Common Fractions, lesson 71. 

Therefore, to find the cube root of a common fraction, 

Find the cube root of the numerator and denominator; thus, 
the cube root of -fc is f, of §£ is f-, &.c. 

However, in many common fractions, like f, £-, &c., 
we cannot get the root of both numerator and denominator 
in whole numbers. In such cases, and in mixed numbers, 
it is best to change the fractions to decimals first, and then 
get the root. 


Lesson 190. 

1. What is the cube root of .000825 ? 

Ans. .0938, nearly. 

2. What is the cube root of 27.98 ? Ans. 3.036, nearly. 

3. What is the cube root of 1,601,618 ? 

Ans. 117, about. 

4. What is the cube root of 2 5 tV 2 t Ans. tV 

5. What is the cube root of fjj ? Ans. .7048, nearly. 

6. What is the cube root of 881^ ? Ans. 9.588, nearly. 

7. If the earth, or globe, be 7,920 miles in diameter, 
how large a cube will it make ? 

Ans. a cube 6,383£ miles square, nearly. 

8. A block of wood, the ends of which are square, and 
the length twice the breadth, or depth, contains 13.718 
cubic ft. ; what is its breadth, or depth, and length ? 

Ans. breadth or depth, 1.9 ft., length 3.8 ft. 

9. I have 81 cubic inches of lead ; if I melt it, how 
large a block will it make, the base of which shall be 
square, and the height f of its length or breadth ? 

Ans. a block 4£ in. square at the base, and 4 in. high. 


How do we get the cube of a -common fraction ? 

How do we find the cube root of a common fraction ? What is 
the cube root of 57 ? Of 1| ? 

What course do we take with mixed numbers; and what with 
common fractions when we cannot get the cube root of both numera¬ 
tor and denominator in whole numbers ? 



SPECIFIC GRAVITY. 


241 

Explanation. The block will contain f as much as a 
cubic block of the same length and breadth. 

10. What is the cube root of .00064 ? Ans. .04. 

11. There is a pile of wood 4 ft. wide, 6 ft. high, and 
21^ ft. long ; how large a cubic pile will it make ? 

Ans. a cubic pile 8 ft. square. 


SPECIFIC GRAVITY. 

Lesson 191. 

The specific gravity of any substance is the ratio of its 
weight to the weight of an equal bulk of fresh water ; thus, 
if the substance weighs 2 oz., and an equal bulk of water 1 
oz., its specific gravity is 2; if it weighs 1 oz., and an 
equal bulk of water 2 oz., its specific gravity is or .5. 
See Promiscuous Questions after Fractions, lesson 85. 

Therefore, to get the specific gravity of any substance, 

Divide its weight by the iveight of an equal bulk of fresh 
water. 

A cubic foot of pure fresh water, properly distilled, 
weighs 1,000 ounces, avoirdupois, at the temperature of 
melting ice. 

The table on the following page shows the specific grav¬ 
ity of several of the most common and important substan¬ 
ces. It is formed by dividing the weight of a cubic foot 
of each substance, in ounces, by 1,000. If we omit the 
point, we get the weight of a cubic foot of each substance 
in ounces. 

Note. It is not necessary to commit the table to memory. 


What is the specific gravity of any substance ? What if the sub¬ 
stance weighs 2 oz., and an equal bulk of water 1 oz. ? What if it 
weighs 1 oz., and an equal bulk of water 2 oz. ? 

How then do we get the specific gravity of any substance ? 

What is said of the weight of a cubic foot of water i 
How is the table formed ? 


21 




242 


SPECIFIC GRAVITY. 


METALS. 

Platina, pure,...... 19.500 

Platina, hammered, . 21.500 
Gold, pure and cast, 19.260 
Gold, hammered,... 19.360 

Mercury, . ..13.560 

Lead, cast,.. 11.350 

Silver, pure and cast, 10.470 
Silver, hammered,.. 10,510 

Copper, cast,. 8.790 

Copper, hammered, 8.890 

Brass, cast, .. 8.400 

Brass, hammered, .. 8.500 

Iron, cast,. 7.210 

Iron, hammered, ... 7.790 

Steel,. 7.840 

Tin, cast,. 7.300 

Zinc, cast,. 7.200 

STONES, EARTHS, &.C. 

Granite,. 2.700 

Marble, . 2.700 

Slate,. 2.700 

Glass,. 2.600 

Flint stone,.. 2.580 

Paving stone,. 2.580 

Freestone,. 2.500 

Clay,. 2.200 

Sand,. 1.500 

Anthracite coal, from 1.400 

to. 2.000 

Bituminous coal, from 1.100 
to. 1.300 


Brick, from 1.800 to 2.000 


woods, &c. 

Lignum Vitse,. 1.300 

Ebony,. 1.200 

Hempen rope,or cable, 1.100 

Mahogany,. 1.000 

Boxwood,. 1.000 

Shell bark hickory, . 1.000 

White oak, heart,.. .930 

Ash,.800 

Rock maple,.760 

White pine,. .570 

Charcoal,.400 

Cork,.240 

LIQUIDS, &C. 

Sulphuric acid, .... 1.840 

Nitric acid,. 1.220 

Sea water,. 1.030 

Cow’s milk,. 1.030 

Pure fresh water, .. 1.000 

Whale oil,.920 

Tallow,.920 

Olive oil, ..910 

Proof spirit,.920 

Alcohol,.840 

GASES. 

Oxygen gas,.00134 

Carbonic acid gas, . .00164 

Common air,.00122 

Nitrogen gas,...... .00098 

Hydrogen gas,.00008 


Note. The specific gravity of any solid, liquid, or gas, increases with 
the cold, and diminishes with the heat. Moreover, there is always 
some difference in the specific gravity of several varieties of the same 
substance. 


1. I have 16 round bars of hammered copper, 12.ft. long 
and of a foot in diameter ; what do they all weigh if we 
reckon the circumference 3 times the diameter ? 

Ans. 800 lbs. 1.6 oz. 
Explanation. Find the weight of a cubic foot in the table. 


What does the specific gravity of any solid, liquid, or gas, increase 
with ? Diminish with ? Is the specific gravity of several varieties of 
the same substance always the same ? 














































SPECIFIC GRAVITY. 


243 


2. A man melted a number of pieces of lead, and cast 

them in a prism 1.5 ft. long, 1 ft. high, and .75 ft. wide ; 
what did it weigh ? Ans. 798 lbs. £ oz. 

3. There is a dish of mercury, or quicksilver, 8 inches 

square on the top, 6 inches square on the bottom, and 4 
inches deep ; how much mercury, by weight, does it con¬ 
tain ? Ans. 96 lbs. 12£ oz., about. 

4. How large a cube will 58,400 ounces of tin make ? 

Ans. a cube 2 ft. square. 

5. There is a monument in a grave-yard composed of 
black marble, in the shape of a cone, 4 ft. in diameter at 
the base, and 7 ft. high ; what is its weight ? 

Ans. 4,948 lbs., about. 

6. What is the weight of a thousand heavy bricks, each 
being 8 inches long, 4 inches wide, and 2 inches thick ? 

Ans. 4,629 lbs. 10 oz., about. 

7. What is the weight of a coil of 5 inch rope, that is, 
of rope 5 inches in circumference, containing 45 fathoms, 
if we reckon the circumference 3 times the diameter ? 

Ans. 268 lbs. 9 oz., nearly. 

8. What is the weight of a ball of white oak 1 ft. in 

diameter ? Ans. 30 lbs. 6^ oz., about. 

9. What is the weight of a barrel of sea water, the 
length being 36 inches, head diameter 18 inches, and 
imng diameter 21 inches, the staves being quite curving ? 

Ans. 421 lbs., about. 

10. There is an iron pipe 5 inches in diameter on the 

outside, and 4 inches on the inside ; how much does a 
piece 100 feet long weigh ? Ans. 2,212 lbs., nearly. 

Lesson 192. 

1. There is a bottle which weighs 6 oz. ; when filled 

with water it weighs 36 oz., and when filled with olive 
oil it weighs 33 oz. 7 T 2 0 drams ; what is the specific grav¬ 
ity of the oil ? Ans. .915. 

2. A bottle weighs 4 oz. when empty, 20 oz. when filled 
with water, and 33 oz. 7 drams when filled with sulphuric 
acid ; what is the specific gravity of the acid ? 

Ans. 1.840, nearly. 

3. There is a piece of dry white pine in the shape of a 

wedge, weighing 35 oz., the length is 1 ft., breadth 6 inches, 
and the thickness of the head 3 inches ; what is its specific 
gravity ? Ans. .560. 


244 


SPECIFIC GRAVITY. 


4. A large glass bottle, which is found to contain 98 
cubic inches of water, weighs 1 lb. when the air is taken 
out of it by an air pump ; when the air is admitted it 
weighs 1 lb. 30 grains, and when filled with hydrogen gas 
it weighs 1 lb. 2 grains ; what then does a cubic foot of 
air weigh, what does a cubic foot of hydrogen gas weigh, 
and what is the specific gravity of air, and of hydrogen 
gas, as found by the experiment ? 

Ans. a cubic ft. of air weighs 529 grs., nearly ; a cubic 
ft. of hydrogen gas weighs 35.27 grs., nearly ; the specific 
gravity of air is .00121, nearly, and of hydrogen gas, 
.00008, about. 

5. There is an ingot composed of gold and copper, the 
specific gravity of which is 17.800 ; in what proportion are 
the gold and copper mixed ? 

Ans. 9,010 of gold, to 1,460 of copper. 

Explanation . See Alligation Alternate. Consider the 
two metals as cast. 

6. I wish to make a gallon of proof spirit from some 
alcohol, or rectified spirit of wine, and some pure water ; 
what quantity of each must I take ? 

Ans. | of a gallon of each. 

7. A man has 16 gallons of French brandy, consisting 

of 10 gallons of alcohol, and 6 gallons of pure water ; 
what is its specific gravity ? Ans. .900. 

8. How much water must be mixed with 4 gallons of 

rum, the specific gravity of which is 890, to reduce it to 
proof spirit ? Ans. 1£ gallon. 

Lesson 193. 

To find the specific gravity of a small irregular shaped 
body, heavier than water. 

First find its weight, then find how much it weighs 
when suspended by a string fastened to one of the scales 
of an accurate balance, the body being immersed in pure 
cold water. In this position it will evidently lose just the 
weight of an equal bulk of water ; because, if it was of the 
same specific gravity as the water, it would be just buoyed 
up by it, and lose all its weight. 

Now to find its specific gravity, 

Divide its weight by the weight lost. 

How do we proceed to find the specific gravity of a small irregufer 
body heavier than water ? 



SPECIFIC GRAVITY. 


245 


To find the specific gravity of a small irregular body 
lighter than water. 

First find its weight, then take a piece of lead, or other 
substance, sufficiently heavy to sink the body in water 
when attached to it ; find how much the lead weighs when 
immersed in water ; then attach the body to it, and im¬ 
mersing both, find the weight, and subtract it from the 
weight of the lead in the water. The difference added to 
the weight of the body evidently gives the weight of an 
equal bulk of water. 

Now to find the specific gravity, 

Divide the weight of the body by the weight of an equal 
bidk of water. 

We can also find the specific gravity of a small irregu¬ 
lar shaped body by dividing its weight by the weight of a 
quantity of water of the same bulk. The bulk can be 
found by immersing it in water, as dire'cted in Mensuration, 
lesson 172. This method is not so accurate as the preced¬ 
ing ones, especially if the body be quite small. 

1. What is the specific gravity of a piece of gold money 
that weighs 216 grains out of the water, and 204 grains 
when immersed ; and if the piece was formed by alloying 
pure gold with pure silver, what quantity of each ingredient 
is there in it ? 

Ans. the specific gravity is 18, and it contains 183 grs. 
of gold, and 33 grs. of silver, about. 

2. A piece of pine charcoal weighs 437.5 grs., and a 

piece of lead weighs, when immersed in water, 956.25 
grs. ; attaching the charcoal to the lead and immersing 
them, they weigh 300 grs. ; what is the specific gravity of 
the charcoal ? Ans. .400. 

3. If a piece of marble weighs 180 pounds, and you find 

the bulk by immersing it in water, as directed in Mensura¬ 
tion, lesson 172, to be l cubic foot, what is its specific 
gravity ? Ans. 2.880. 


How do wo proceed to find the specific gravity of a small irregular 
body lighter than water ? 

How, also, can we find the specific gravity of a small irregular body ’ 
What is said of the accuracy of this method ? 

21* 



246 


SIMPLE MACHINES. 


SIMPLE MACHINES, 

OFTEN CALLED 
MECHANICAL POWERS. 


Lesson 194 . 

There are usually reckoned six simple machines ; the 
lever, the wheel and axle, the pulley, the inclined plane, the 
screw, and the wedge. 

The force that raises a weight, or overcomes a resist¬ 
ance, is called the power. The power is usually the force 
of men, oxen, horses, moving water, wind, &c. 


THE LEVER. 


The support or prop round which the lever moves, is 
called the fulcrum. 


Figure 22. 




There are three kinds of 
levers. In the first kind the 
fulcrum is between the power 
and the weight, as in figure 
22 . 


In the second kind the 
weight is between the power 
and the fulcrum, as in figure 
23 . 


How many simple machines are there usually reckoned ? Name 
them. 

What is called the power? What is usually the power ? 

What is called the fulcrum? 

How many kinds of levers are there ? Explain them ? 












THE LEVER. 


247 


In the third kind the power 
is between the weight and ful¬ 
crum, as in figure 24. 

When these levers are 
extremely light, experiment 
shows that in order to sustain 
a weight with either of them, 

The power must be to the weight in the same proportion as 
the distance from the fulcrum to the weight is to the distance 
from the fulcrum to the power . 

To be performed in the mind. 

1. In what proportion must the power be to the weight 
with the first kind of lever, in order to balance the weight, 
if the fulcrum is placed in the centre of the lever ? If the 
weight be 10 lbs., how many pounds must the power be 
equal to ? 

2. A lever of the second kind is 6 ft. long, and the weight 
is placed 3 ft. from the fulcrum ; in what proportion must 
the power be to the weight in order to sustain it ? If the 
weight be 12 lbs., how much must the power lift ? 

3. A lever of the third kind is 8 ft. long, and the power 
is applied 4 ft. from the fulcrum ; in what proportion must 
the power be to the weight to sustain it ? If the weight 
be 13 lbs., how much must the power lift ? 

4. A lever of the first kind is 7 ft. long, and the fulcrum 
is 2 ft. from the weight ; what proportion must the power 
bear to the weight to balance it ? 

5. A lever of the second kind is 10 ft. long, and the 
weight is placed 3 ft. from the fulcrum ; in what proportion 
must the power be to the weight in order to sustain it ? 

6. A lever of the third kind is 3 ft. long, and the power 
is applied 4 ft. from the fulcruin ; in what proportion must 
the power be to the weight in order to sustain it ? 

Lesson 195 . 

To be performed in the mind. 

1. A lever of the first kind is 6 ft. long, and the fulcrum 
is 2 ft. from the weight ; what proportion must the power 


Figure 24. 




When these levers are extremely light, what does experiment show ? 






248 


THE LEVER. 


bear to the weight in order to balance it ? What part of 
the weight must the power be ? If the weight be 42 lbs., 
how many pounds must the power be equal to ? If the 
power be 9 pounds, what weight will it balance ? 

2. A lever of the second kind is 15 ft. long, and the weight 
is 5 ft. from the power ; in what proportion must the power 
be to the weight to sustain it ? What part of the weight 
must the power be ? If the weight be 60 lbs., how many 
pounds must the power lift ? If the power lift 12 lbs., 
how many pounds will it sustain ? 

3. A lever of the third kind is 12 ft. long, and the power 
is 9 ft. from the fulcrum ; what proportion must the power 
bear to the weight'in order to sustain it ? What part of 
the weight must the power be ? If the weight be 72 lbs.', 
how many pounds must the power lift ? If the power be 
12 lbs., what weight will it sustain ? 

4. There is a lever of the first kind, 8 ft. long ; what 
weight will 12 lbs. as a power, on the end of it, balance, 
if the fulcrum be placed 2 ft. from the weight ? 

5. A lever of the second kind is 9 ft. long, and a power 
can be applied that will lift 10 lbs. ; what weight, placed 
3 ft. from the fulcrum, will the power sustain ? 

6. A lever of the third kind is 7 ft. long, and a power 
can be applied 4 ft. from the fulcrum, that will lift 14 lbs.; 
what weight placed at the end of the lever will it sustain ? 

7. There is a lever of the first kind 6 ft. long, the power 
is equal to 10 lbs., and the weight is 30 lbs. ; where must 
the fulcrum be placed, so that the power shall balance the 
weight ? 

8. A lever of the second kind is 12 ft. long, the power 
amounts to 7 lbs., and the weight is 28 lbs. ; where must 
the weight be placed, so that the power shall sustain it ? 

9. A lever of the third kind is 8 ft. long, the power 
amounts to 16 lbs., and the .weight is 14 lbs. ; where must 
the power be placed, so as to sustain the weight ? 

10. If the weight is suspended 1 inch from the fulcrum 

in a steelyard, at what distance from the fulcrum must a 
poise weighing 1 lb. be placed, to balance a weight of 1 
lb. ? A weight of 2 lbs. ? 2£ lbs. ? 3 lbs. ? 11 lbs. ? 

If the first notch is 1 inch from the fulcrum, and the 
notches are tV of an inch apart, how many pounds and 
ounces will the poise balance, when placed in the 2d 
notch ? 3d notch ? 4th notch ? 5th notch ? 8th notch ? 
16th notch ? 64th notch ? 72d notch ? 80th notch ? 


THE LEVER. 


249 


Lesson 196. 

For the Slate. 

1. A weight of 2,000 lbs. is on the end of a lever of the 

first kind, 1 ft. from the fulcrum; how many pounds as a 
power must be placed 10 ft. from the fulcrum to balance 
the weight i Ans. 200. 

2. A lever of the second kind being 12 ft. long, where 
must a weight of 720 lbs. be placed, so that a power capa¬ 
ble of lifting 50 lbs. shall sustain it ? 

Ans. 10 in. from the fulcrum. 

3. A lever of the third kind is 15 ft. long, with a weight 

at the end of 90 lbs. ; if a power be applied 14 ft. from the 
weight, how many pounds must it be capable of lifting in 
order to sustain the weight ? Ans. 100. 

4. A lever of the first kind, 11 ft. long, has a weight of 
1,500 lbs. at one end, and a power equal to 150 lbs. can be 
applied at the other ; where must the fulcrum be placed so 
that the power shall balance the weight ? 

Ans. 1 ft. from the end. 

5. I wish to sustain 933^ lbs. on a lever of the second 
kind, 14 ft. long, the fulcrum of which is 3 ft. from the 
weight; what power must I employ ? 

Ans. a power capable of raising 200 lbs 

6 . Ifyou have a lever of the third kind 10 ft. long, with 
a weight at the end of 145 lbs., where must you apply a 
power equal to 900 lbs. to sustain the weight ? 

Ans. 1 ft. 74 in. from the fulcrum. 

7. If the place of suspending the weight in a steelyard 
be 2 inches from the fulcrum, what weight will a poise 
weighing 4 lbs. balance, if placed 3 ft. from the fulcrum ? 

Ans. 72 lbs. 

8. The place of suspending the weight in a steelyard be¬ 

ing 1£ inches from the fulcrum, at what distance from the 
fulcrum must a poise weighing 4 lbs. be placed, so as to 
balance a weight of 56 lbs. ? Ans. 1 ft. 9 in. 

Observations on the Lever. We have regarded the lever 
as very light ; the weight of it, however, in practice, will 
increase or diminish the power a little. By examining 
figures 22, 23, and 24, lesson 194, we see that the weight 
of the lever of the first kind, will generally increase the 


What will the weight of the lever do in practice ? What do we see 
by examining figures 22, 23, and 24, lesson 194 ? 



250 


THE WHEEL AND AXLE. 


power, and that the weight of the levers of the second and 
third kinds, will generally diminish the power. 

When the power and weight balance each other, the 
power must be increased a little in order to raise the 
weight, as there is some rubbing or friction on the fulcrum. 

The lever is a very important instrument, and is used in 
a great many different forms ; when you employ your 
weight on the end of an iron bar to overturn a log, the bar 
is a lever of the first kind ; but if you place your shoulder 
beneath the bar to roll over the log, the bar is a lever of 
the second kind. A man who loads hay with a pitchfork, 
employs a lever of the third kind, one hand being the 
fulcrum, and the other the power. Common tongs are 
double levers of the third kind ; blacksmiths’ tongs are 
double levers of the first kind, the fulcrum being at the 
pivot. 

Lesson 197. 

THE WHEEL AND AXLE^ 

The principle of 
the lever of the 
first kind, is em¬ 
ployed in the wheel 
and axle ; for the 
pivot, or axis, acts 
as a fulcrum, the 
radius of the axle 
is the distance of 
the weight from 
the fulcrum, and 
the radius of the 
wheel is the dis¬ 
tance of the power 
from the fulcrum. 

Therefore, to balance the weight, 

The power must be to the weight as the radius of the axle 
is to the radius of the wheel . 

When the power and weight balance each other, what must be done 
in order to move the weight ? Why ? 

What is an example of a lever* of the first kind ? Second kind ? 
Third kind ? What are common tongs ? Blacksmith’s tongs ; and 
where is the fulcrum ? 

What principle is employed in the wheel and axle ? Explain how. 

How must the power be to the weight in order to balance it ? 




















THE WHEEL AND AXLE. 


251 


1. What power must be applied to the circumference of 
tt wheel 6 ft. in diameter, to balance 350 lbs., suspended 
from the axle, which is .5 of a foot in diameter ? 

Ans. 29 lbs. 2§ oz. 

2. The diameter of the wheel being 7 ft., and that of the 

axle being 9 in., what weight will a power equal to 12 lbs. 
balance ? Ans. 112 lbs 

3. What must the diameter of a wheel be so that a 

power equal to 40 lbs. may balance a weight of 500 lbs., the 
axle being 1 ft. in diameter ? Ans. 12£ ft. 

4. What must the diameter of the axle be, so that a 

power equal to 30 lbs. may balance a weight of 270 lbs., 
the wheel being 9 ft. in diameter ? Ans. 1 ft. 

5. There are 2 axles, side by side, 1 ft. in diameter, 
each of which has a wheel 10 ft. in diameter ; a belt pass¬ 
es round the first axle and the second wheel ; how many 

pounds on the second axle, will a power equal to 25 lbs. on 
the first wheel balance ? Ans. 2,500 lbs. 

6. A wheel is 9 ft. in diameter, and the axle 10.8 in., the 

power is equal to 38 lbs., and‘the weight is 310 lbs.; 
which will overcome in this machine, the weight or the 
power ? _ Ans. The power. 

Observations on the Wheel and Axle. When the power and 
weight balance each other, the power must be increased a 
little in order to overcome the friction on the axis, and 
raise the weight ; the power may also be diminished a little 
before the weight will overcome the friction and descend. 

The wheel and axle are used in a great many different 
forms ; the crank of the windlass with which water is 
sometimes drawn, turns round in a circle, and acts as a 
wheel, while the body of the windlass acts as an axle ; the 
windlass of a vessel acts as a wheel and axle ; the capstan 
of a vessel is, in effect, a wheel and axle, where the wheel, 
represented by the levers, is horizontal, &.c. 


What is said of friction in the wheel and axle ? 

Name some of the forms in which the wheel and axle are used ? 



252 


THE PULLEY. 


Lesson 198. 



A pulley with an immovable block, like figure 26, is call¬ 
ed a fixed pulley. A pulley with a movable block, like 
figure 27, is called a movable pulley. With one fixed pulley, 
like figure 26, the power must evidently be equal to the 
weight in order to balance it ; this pulley merely serves to 
change the direction in which the power operates. With 
one movable pulley, like figure 27, the weight must be 2 
times the power in order to be balanced, for the weight is 


What is called a fixed pulley ? A movable pulley ? With one fixed 
pulley, how is the power to the weight P What does the fixed pulley 
serve for ? With one movable pulley, how is the power to the weight ? 




























THE PULLEY. 


253 


supported by 2 ropes. In figure 28, we see that the 
weight is supported by 4 ropes, and is evidently 4 times the 
power required to balance it. In figure 29, we see that the 
weight is supported by 5 ropes, and is evidently 5 times the 
power required to balance it. 

Therefore, to find what weight the power will balance, 

Multiply it by as many ropes as support the weight. 

1. What weight will a power equal to 80 lbs. balance 

by means of pulleys, the weight being supported by 4 
ropes ? Ans. 320 lbs. 

2 . There are 3 movable pulleys, and one end of the rope 
is attached to the block which contains them ; what power 
will be necessary to balance a weight of 420 lbs. 

Ans. 60 lbs. 

3. A system of pulleys support the weight by means of 8 

ropes, and the power is applied through an axle, .8 of a 
foot in diameter, to which there is a wheel 8 feet in diame¬ 
ter ; what power at the wheel will balance a weight of 
5,000 lbs. Ans. 62£ lbs. 

Observations cn the Pulley. When the power and weight 
balance each other, the power must be increased in order 
to raise the weight, £ of the power being usually lost in 
overcoming friction ; the power may also be diminished 
about £ before the weight will overcome the friction and 
descend. 

4. What power operating on pulleys will be necessary 
to raise a weight of 360 lbs. supported by 6 ropes ? 

Ans. 90 lbs. 

5. What weight will a power equal to 50 lbs. raise by 
means of 2 movable pulleys, one end of the rope being 
fastened to the block which contains them ? 

Ans. 166§. lbs. 

6 . There is a power equal to 30 lbs. and a weight of 
200 lbs. ; if one end of the rope is fastened to the block of 
the fixed pulleys, how many movable pulleys must 
there be so that the power shall raise the weight ? 

Ans. 5. 


What do we see in figure 28, lesson 198? In figure 29, lesson 
198? 

What is the rule to find what weight the power will balance ? 

What is said of friction in the pulley ? 

22 



254 


THE INCLINED PLANE. 


Lesson 199. 


THE INCLINED PLANE. 



Figure 30. 


If we place a very 
smooth body on an 
inclined plane, very 
smooth and slippery, 
we find that to hin¬ 
der the body from 
moving down the 
plane, 


A power must be employed , parallel to the plane , which 
shall be to the weight of the body as the height of the plane 
is to its length. 

1 . There is an inclined plane 8 ft. long, and 3 ft. high *„ 

what weight will be balanced on it by a stone weighing 12 
lbs., acting by means of a line over a fixed pulley, at the 
top of the plane ? Ans. 32 lbs. 

2 . What power will be necessary to hinder an iron 
cylinder, weighing 600 lbs., from rolling down a plane; the 
length of which is 25 ft., and the height 5 ft. ? 

Ans. 120 lbs. 

3. There is a platform 39 ft. long, one end of which is 
to be raised so that a power equal to 100 lbs. shall just 
sustain a wagon weighing 975 lbs. on it ; how high must 


the end be raised ? 


Ans. 4 ft. 


4. There is an inclined plane, 90 ft. long, and 12 ft. high, 
on which is a wagon weighing, with its load, 6,000 lbs. ; 
what power applied through a system of pulleys support¬ 
ing the weight by 8 ropes, will sustain the wagon on the 


plane ? 


Ans. 100 lbs. 


Observations on the Inclined Plane. When the power 
and weight balance each other, the power must be increas¬ 
ed a little, in order to overcome the friction, and cause 


If we place a very smooth body on an inclined plane, very smooth 
and slippery, what do we find ? 

What is said of friction on an inclined plane ? 




THE SCREW. 


256 

the weight to move up the plane. The power may also 
be diminished a little, before the weight will overcome the 
friction and descend. 


Lesson 200. 

THE SCREW. 


Figure 31. 



The screw consists of two parts, the external screw and 
the internal screw. The external screw is a cylinder with 
a spiral projection, called the thread , coiled round it. This 
thread fits into a groove, cut in the internal screw. The 
thread is an inclined plane, and operates like one. It 
rises in passing round the screw the distance between the 
tops of the coils or turns. When the thread is triangular, 
this distance is evidently the same as that between the 
coils. Therefore, in order that the screw may balance the 
weight and sustain it without turning, we must apply to the 
thread a power which shall be to the weight as the dis¬ 
tance between the tops of the coils is to the length of one 
turn or revolution of the thread. Now if the power is ap¬ 
plied at the end of a lever, we may imagine the thread to 
extend to the end of the lever, and then it is obvious that 


What does the screw consist of ? What is the external screw ? Into 
what does this thread fit? What is the thread, and how does it operate? 
How much does it rise in passing round the screw? When the thread 
is triangular, what is this distance the same as ? What is necessary 
in order that a screw may balance the weight and sustain it without 
turning ? 










256 


THE SCREW. 


To balance the weight, 

The power must be to the weight as the distance between 
the lops of the coils is to the circumference described bp the 
lever. 

1. A certain screw has a triangular thread, the coils of 
which are 1 inch apart; what weight will be balanced on it 
by a power equal to 40 lbs., applied at the end of a lever 
4 ft. from the centre of the screw ? 

Ans. 12,063.7 lbs., about. 

Note. The circumference described by the lever, in example 1 , is a 
mere trifle more than the circumference of a circle of which the lever 
is the radius, because the lever rises 1 inch, in turning round. 

2 . If a screw has a square thread with the tops of the 

coils of an inch apart, and a lever 36 in. long, what 
power applied to the lever will balance a weight of 1,500 
lbs. ? Ans. 5 lbs. 4^ oz., about. 

3. There is a weight of 7,540 lbs. to be sustained by a 
screw, to which a power equal to 25 lbs. can be applied 
by means of a lever 6 ft. long ; the thread being square, 
what distance must the tops of the coils be apart ? 

Ans. 1£ inch, nearly. 

4. There is a downward pressure of 5,000 lbs. to be ex¬ 
erted by a power equal to 40 lbs., through the medium of a 
screw having a triangular thread the coils of which are 

of a ft. apart ; how long must the lever be ? 

Ans. 3.18 ft., about. 

Observations on the Screw. When the power balances 
the weight, the power must be increased in order to raise 
the weight ;'■£ of the power being usually lost in overcom¬ 
ing friction. The power may also be diminished about ^ 
before the weight will overcome the friction and descend. 

5. There is a weight of 1,508 lbs. to be raised 5 ft. high 

by means of a screw which has a lever 5 ft. long, and a 
square thread with the tops of the coils £ of an inch apart; 
what power will be necessary ? Ans. 4^ lbs. 

6 . What weight will a power equal to 35 lbs. raise by 
means of a screw with a lever 4 ft. long, the thread being 
triangular, and the coils 1^ inch apart ? 

Ans. 5,277.9 lbs., nearly* 

If the power is applied at the end of a lever, what is necessary to bal¬ 
ance the weight ? 

What is said of the friction of the screw ? 




THE WEDGE. 


257 


Lesson 201. 

THE WEDGE. 

Figure 32. 

If you hold the weight in figure 
30 firmly, and raise it by pushing 
along the inclined plane, the plane 
becomes a wedge ; now to balance 
the weight, the power must evi¬ 
dently be to the weight as before, 
that is, as the height of the plane is 
to its length, or as the thickness of 
the head of the wedge is to the 
length of the slanting side. The 
only difference will be in the greater 
friction, and difficulty in moving 
the plane or wedge instead of the 
weight. A common wedge is two 
inclined planes placed together, as the dotted line in figure 
32 represents. 

Therefore, to balance the resistance, 

The power must be to the resistance, as the thickness ; of the 
head is to the length of both the slanting sides. 

1 . There is a wedge with a head 2 in. thick, the length 
of each side being 10 in. ; what amount of resistance will 
a power equal to 50 lbs. applied to the head balance ? 

Ans. 500 lbs. 

2 . What power applied to a wedge 3 in. thick at the 

head, and 12 in. long on one side, will balance a resistance 
equal to 1,800 lbs. ? Ans. 225 lbs. 

3. A power equal to 80 lbs. is to be exerted on a wedge 

to balance a resistance amounting to 1,200 lbs. ; what 
must the thickness of the head be, the length of a side 
measuring 15 in. ? Ans. 2 in. 

Observations on the Wedge. Friction greatly modifies the 
operation of the wedge ; thus, if the power be a weight 



Explain how the inclined plane in figure 30 becomes a wedge. Now 
to balance the weight, how must the power evidently be to the weight ? 
What will the only difference be ? YVhat is a common wedge ? 

To balance the resistance, Ir w must the power be to the resistance? 

v C)Q # 











258 


THE WEDGE. 


placed on the head of a common wedge, it often will not 
overcome part as much resistance'as we should expect 
from the preceding proportion, whereas if the wedge be 
struck, it many times will overcome 10 times this resist¬ 
ance. 

The operation of an ordinary wedge, therefore, can only 
be estimated or guessed at by those accustomed to its use. 
The principle of the wedge, however, is employed in many 
machines, like presses used in printing and manufacturing, 
in which friction operates regularly, and diminishes the 
effect of the power about 

By different combinations of these simple machines, all 
others, however complicated, are formed ; and the preced¬ 
ing rules enable us to calculate the effect of any machine 
whatever. 

There is no power gained by using these simple ma¬ 
chines, or any machine ; on the contrary, some of the 
power is always lost ; still, they enable us to perform what 
would be difficult or impossible from a direct application of 
the power. For instance, a man cannot lift a millstone, 
but by working 20 minutes with a screw and lever, he can 
raise it several feet ; the sum of his exertions, however, 
is about £ greater than the power employed by 20 men who 
lift it the same height in 1 minute, because £ of his labor 
is expended in overcoming friction. 


What thing 1 greatly modifies the operation of the wedge ? Explain 
this modification. 

How can the operation of an ordinary wedge, therefore, be obtained? 
In what is the principle of the wedge employed ? How does friction 
operate in them ? 

What are formed by different combinations of these simple machines ? 
What do the preceding rules enable us to calculate ? 

Is there any power gained by the employment of these simple ma¬ 
chines, or of any machine ? What on the contrary is true ? What do 
they enable us to perform? Give an instance. What is said of the sum 
of his exertions ? 




PROMISCUOUS QUESTIONS. 


259 


PROMISCUOUS QUESTIONS 

IN 

MENSURATION, SQUARE ROOT, CUBE ROOT, &c. 

Lesson 202. 

To be performed in the mind. 

I. 729 cubic ft. of clay were dug out of a cubic pit ; 
wha was the depth of the pit ? 

2 How many square yards of carpeting will cover a 
floor 20 ft. long and 18 ft. wide ? 

3. I bought 8,100 sq. ft. of land, to be laid out in an 
exact square ; how long will one side be ? 

4. A lever of the first kind is 12 ft. long, and the fulcrum 
is 2 ft. from the weight ; what power applied to the lever, 
will balance a weight of 60 lbs. ? 

5. A weight of 100 lbs. is to be raised by a wheel and 
axle, the axle being 6 in. in diameter ; what must the di¬ 
ameter of the wheel be so that a power equal to 10 lbs. 
shall balance the weight ? 

6 . What weight can I raise by means of pulleys with a 
power of 40 lbs., the weight being supported by 6 ropes ? 

7. What power will balance an iron cylinder weighing 
70 lbs. on an inclined plane, the length of which is 12 ft., 
and the height 3 ft. ? 

8 . There is a circle 12 ft. in diameter ; if you reckon 
the circumference as 3 times the diameter, how many 
square feet will it contain ? 

9 . A marble monument in the shape of a pyramid is 2 fit. 
square at the base, and 6 ft. high ; how many cubic feet 
does it contain ? 

10 . How many square rods are there in a triangular 
piece of land, the base of which is 10 rods, and the altitude 
8 rods ? 

II . What is the square root of 400 ? 

12 . What is the cube root of 125,000 ? 

Lesson 203. 

1 . There is a block of granite in the shape of a frus¬ 
tum of a square pyramid, the top is 2 ft. square, the 
base 5 ft. square, and the height is 4 ft. ; what is its 
weight ? Ans. 8,775 lbs 


260 


PROMISCUOUS QUESTIONS. 


2 . What is the cube root of 74,088 ? Ans. 42. 

3. There is a house 33 ft. long, 24 ft. wide, 19 ft. from 

the sill to the eaves, and 30 ft. from the sill to the top of 
the roof; how much will it cost to paint the outside at 25 
cts. a sq. yd., no allowance being made for windows and 
doors, ? Ans. $67^r. 

4. I own a piece of ground 250 ft. long, the widths of 

which, measured every 50 ft., are as follows; 35 ft., 18 ft., 
16 ft., 24 ft., 10 ft., and 20 ft. ; how many square feet of 
land are there in the piece ? Ans. 4,775 sq. ft. 

5. The quantity of water flowing over a mill dam, July 

1 st, was found to be 28 cubic ft. a second at 6 o’clock in 
the morning, 23 cubic ft. at 9 o’clock, 19 cubic ft. at 12 
o’clock, 18 cubic ft. at 3 o’clock, and 22 cubic ft. at 6 
o’clock in the evening ; what quantity, on an average, 
passed over the dam in a second ? Ans. 21^ cubic ft. 

6 . There is a square garden containing 2,500 sq. ft. ; 
how far is it between the opposite corners ? 

Ans. 70.71 ft., or 70 ft. 8£ in., about. 

7. How many feet, board measure, are there in a stick 
of timber 2 ft. square, and 22 ft. 9 in. long ? Ans. 1,092. 

8 . What is the surface of a circle 15 ft. in diameter ? 

Ans. 176.714 sq. ft., about. 

9. There is a pine log 4 ft. in diameter, and 27 ft. long, 
the ends ©f which are cut off obliquely, but parallel to each 
other ; how many cubic ft. does it contain ? 

Ans. 339.2ft, about. 

10. How many square inches of leather will cover a foot 

ball 6 in. in diameter ? Ans. 113 T V sq. in., nearly. 

Lesson 204. 

1 . How large a cube will 90,800 ounces of lead make ? 

Ans. A cube 2 ft. square. 

2. How many gallons of milk are there in a cask of 
which the length is 26 in., the head diameter 17 in., and 
the bung diameter 22 in., the staves curving but little ? 

Ans. 29 gals., nearly. 

3. What is the carpenters’ tonnage of a single decked 
vessel, the keel measuring 60 ft., the breadth of the beam 
20 ft., and the depth of the hold 9^ ft. ? Ans. 120 tons. 

4. There is a weight of 730 lbs. to be raised by a lever 
of the first kind 10 ft. long, to which a power can be 
applied equal to 146 lbs. ; how far must the fulcrum be 


PROGRESSION BY DIFFERENCE. 


261 


placet! from the weight so that the power may balance 
it ? Ans. 1 ft. 8 in. 

5. A railroad has an inclined plane 2,000 ft. long, the 
height of which is 250 ft. ; what power must be applied by 
means of an engine at the top of the plane, to pull up a 
train of cars weighing 15,000 lbs., £ of the power being 
expended in overcoming friction ? 

Ans. a power capable of raising 2,812^ lbs. 

6. There is a steelyard the fulcrum of which is 1 in. 

from the place of suspending the weight ; how far from 
the fulcrum must a poise weighing 2 lbs. be placed to bal¬ 
ance a weight of 3 lbs. ? Ans. 1£ in. 

7. There is a wheel 7£ ft. in diameter, the axle of 
which is 9 in. in diameter ; how large a weight will a 
power equal to 100 lbs. applied to the wheel balance ? 

Ans. 1,000 lbs. 

8. There are 2 pulleys in a movable block, and one end 

of the rope is fastened to the block of the fixed pulleys ; 
how large a weight can be raised with this machine by a 
power equal to 96 lbs., \ of the power being lost by fric¬ 
tion ? ' Ans. 288 lbs. 

9. There is a screw with a lever 6 ft. long, measuring 
from the centre of the screw, the thread is square, and the 
tops of the coils are £ of an inch apart ; what power will 
be sufficient to balance a weight of 679 lbs. ? 

Ans. f of a lb., about. 

10 . In the preceding question, what power will be suf¬ 
ficient to raise the weight ? Ans. lb., about. 


PROGRESSION BY DIFFERENCE, 


OFTEN CALLED ARITHMETICAL PROGRESSION. 

Lesson 205. 

A row or series of numbers, constantly increasing or de¬ 
creasing, so that the difference between two adjacent num¬ 
bers is the same in all parts of the row or series, is called 


What is called a progression by difference? 




PROGRESSION BY DIFFERENCE. 


262 

a progression by difference ; thus, 2, 4, 6, 8, 10, 12, is an 
increasing progression by difference, and 16, 11, 6, 1, is a 
decreasing progression by difference. 

The numbers which form a progression by difference, 
are called the terms . 

The difference between any two terms, is called the 
common difference. 

There are five things in progression by difference, any 
three of which being known, the other two may be found ; 
these things are, 

1 st, The first term. 

2 d, The last term. 

3d, The number of terms. 

4th, The common difference. 

5th, The sum of all the terms. 

By examining these five things, we see that three dif¬ 
ferent things can be selected in ten different ways. Now 
in each way three things are chosen, there are two to be 
found, so there are twenty different calculations to make 
in order to obtain the answer to every kind of question that 
may arise in progression by difference. 


We shall now point out the method of obtaining the an¬ 
swers to the most important questions. 


1 . A bookseller has 100 books on his counter ; the first 
is worth 5 cts., the second 8 cts., the third 11 cts., and so 
on ; each being worth 3 cts. more than the one next pre¬ 
ceding ; what is the value of the last book ? 


OPERATION. 

99 

3 

297 

5 cts. value of first book. 

302 cts. value of the last 
book. Ans. 


Explanation. The most ob¬ 
vious way to obtain the value 
of the last book, is to add 3 
cts. to 5 cts., then 3 cts. to 
the result, and so on, adding 
3 cts. to the value of every 
book except the last. It is 
plain, then, that the answer 
can be more readily found by 
adding to 5 cts. 99 times 3 cts. 


Give an example of an increasing and of a decreasing progression by 
difference ? 

What are called the terms ? 

What is called the common difference P 

How many things are there in progression by difference ? How many 
must be known to find the others? Name these things ? 

What do we see by examining these five things ? 

Explain how example 1, lesson 205, is performed 



PROGRESSION BY DIFFERENCE. 


263 


2. A bookseller has 80 books on his counter ; the first is 
worth 162 cts., the second 160 cts., the third 158 cts., and 
so on, each being worth 2 cts. less than the preceding one; 
what is the value of the last book ? 


OPERATION. 

79 

2 

158 

162 cts. value of first book. 
158 

4 cts. value of the last 
book. Ans. 


Explanation. .The most ob¬ 
vious way to obtain the value 
of the last book, is to sub¬ 
tract 2 cts. from 162 cts., 
then 2 cts. from the remain¬ 
der, and so on, taking 2 cts. 
from the value of every book 
except the last. It is plain, 
then, that the answer can be 
more readily found by sub¬ 
tracting from 162 cts. 79 times 
2 cts. 


Therefore, to find the last term, when the first term, 
number of terms, and common difference are known, 

Multiply 1 less than the number of terms by the common 
difference , and add the product to the first term, if it is an in¬ 
creasing progression, but subtract the product from the frst 
term , if it is a decreasing progression. 

3. What is the first term of an increasing progression 

by difference, the last term being 266, the common dif¬ 
ference 6, and the number of terms 40 ? Ans. 32. 

Explanation. Call the first term the last, and the last 
term the first. 

4. Suppose that an empty bottle, on being put 1 ft. be¬ 
neath the surface of the ocean, is pressed on all sides by 
the water to the amount of 50 lbs., and that it sustains an 
additional pressure of 50 lbs. for every foot it sinks; what 
pressure will it bear 501 ft. beneath the surface ? 

Ans. 25,050 lbs. 

5. A man bought 34 yds. of cloth, and agreed to give 

12 cts. for the first yard, 12£ cts. for the second yard, 12f 
cts. for the third yard, and so on ; what did the last yard 
cost him ? , Ans. 23 cts. 


Explain how example 2, lesson 205, is performed. 

How do we find the last term, when the first term, number of terms, 
and common difference are known ? 



264 


PROGRESSION BY DIFFERENCE. 


Lesson 206 . 


1. A teacher gave the scholar that stood at the head of 
his class 32 chestnuts, the next 29, the next 26, and so on 
to the last, who received only 2 ; how many scholars were 
there in the class ? 

OPERATION. 

32 number the first receives. 

2 number the last receives. 


Common Q \ or) 
difference. ^ 


10 

1 


11 scholars. Ans. 


then count the number of shares. 


Explanation. 
The most obvious 
way to obtain the 
answer, is to take 
3 chestnuts from 
32, then 3 from 
the remainder, 
and so on, taking 
3 from the share 
of every one ex¬ 
cept the last, and 
But the answer can 


be more readily found ; for if we subtract 2 from 32, the 
remainder, 30, is as many times 3 as there are shares ex¬ 
cept the last ; so by dividing 30 by 3, and adding 1 to the 
quotient, we obtain the answer. 

Therefore, to find the number of terms, when the first 
term, last term, and common difference are known, 


Divide the difference between the first and last terms by the 
common difference , and add 1 to the quotient. 

2. The first term of a progression by difference being 8, 

the last 100, and the common difference 4, what is the 
number of terms ? Ans. 24. 

3. Experiment shows us that a heavy body falls through 
the air 16 ft. the first second, 48 ft. the next second, 80 ft. 
the third second, and so on ; now how many seconds has a 
stone been falling that descends 336 ft. in a second ? 

Ans. 11. 

4. A young man having 50 cts., determined to add 6£ 

cts. to it every day ; after some time, on counting his 
money, he found he had $13 ; how many days had he ad¬ 
hered to his resolution ? Ans. 200. 

Explanation. 50 cts. is one term ; so there is 1 term 
more than the number of days. 


Explain how example 1, lesson 206, is performed. 

How do we find the number of terms when the first term, last term, 
and common difference are known ? 



PROGRESSION BY DIFFERENCE. 


265 


5. There are 21 cannon balls of different weights lying 
in a heap, the smallest weighs 4 lbs., and the largest 44 
lbs., and there is a regular increase of weight from the 
smallest to the largest ; what is this increase ? 


21 

1 


OPERATION. 

44 

4 


20 210 ) 410 


last ; so 


2 lbs. 
we divide 


Ans. 

the whole 


obtain what'each ball increases. 


Explanation. The dif¬ 
ference between 4 lbs. and 
44 lbs., is the whole in¬ 
crease of weight ; now the 
regular increase is added 
to every ball except the 
increase, 40 lbs., by 20 to 


Therefore, to find the common difference when the first 
term, last term, and number of terms are known, 

Divide the difference betioeen the first and last terms by 1 
less than the number of terms. 

6. The first term of a progression by difference being 

122, the last term 7, and the number of terms 24, what is 
the common difference ? Ans. 5. 

7. A man’s expenses for the first year after marriage 

were $180, but they gradually increased for 29 years, 
when they were $2,000 a year ; what was the yearly in¬ 
crease ? Ans. $65. 

8. The weight of 5 men forms a progression by differ¬ 
ence, the first man weighing 135 lbs., and the last 239 lbs. ; 
what is the common difference of weight between them ? 

Ans. 26 lbs. 


Lesson 207. 

1. A grocer bought 7 barrels of flour at different times ; 
the first barrel cost $4, and the price of each succeeding 
barrel increased regularly till the seventh cost $16 ; how 
much did the whole cost ? 


Explain how example 5, lesson 206, is performed. 

How do we find the common difference, when the first term, last 
term, and number of terms are known ? 

23 



266 


PROGRESSION BY DIFFERENCE. 


OPERATION. 

4 

16 

2)20 

10 

7 


Explanation. As the price regularly 
increases from the first barrel to the last, 
it is obvious that the average price of 
the first and last barrels, is the average 
price of each. We multiply the average 
of $4 and $16, then, by 7 for the an¬ 
swer. 


$70 Ans. 

Therefore, to find the sum of all the terms, when the 
first term, the last term, and the number of terms are 
known, 

Multiply the average of the first and last terms by the num¬ 
ber of terms. 

2. There is a progression by difference, the first term of 

which is 5, the last 595, and the number of terms is 60 ; 
what is the sum of all the terms ? Ans. 18,000. 

3. If a stone has been falling 14 seconds, and moves 

464 ft. in a second, from what height has it fallen, the de¬ 
scent in the first second being 16 ft. ? Ans. 3,360 ft. 

4. There are a number of rows of corn, the first of 

which contains 3 hills, the second 7, the third 11, and so 
on to the last, which has 43 hills ; how many hills are 
there in all the rows ? Ans. 253. 

Explanation. What is the number of rows ? 

Note. We have given no rules to find the answers.to many of the 
twenty different questions in progression by difference; the answers, 
however, can usually be obtained by means of what precedes, some¬ 
times indirectly, that is, after finding something else first, as in the pre¬ 
ceding question. 

5. A man paid $2.20 for a number of small books, the 

prices of which formed a progression by difference ; the 
price of the first was 5 cts., and the price of the last 35 
cts. ; how many books did he buy ? Ans. 11. 

Explanation. Examine the last rule. 


Explain how example 1, lesson 207, is performed. 

How do we find the sum of all the terms, when the first term, the 
last term, and the number of terms are known ? 

What have we given no rules for ? How can the answers be usually 
obtained ? 


/ 




PROGRESSION BY QUOTIENT. 


267 


6 . In 9 seconds, an iron ball descended 1,296 ft., falling 

272 ft. the last second ; how far did it fall the first sec¬ 
ond ? Ans. 16 ft. 

7. Suppose a number of soldiers to be formed into a body 
resembling a wedge, having 7 men in the first rank, 9 in 
the second, 11 in the third, and so on, till there are 25 
ranks ; how many soldiers will there be in the whole body ? 

Ans. 775. 

8 . If a body of soldiers drawn up like the preceding had 

4 men in the first rank, 8 in the second, 12 in the third, 
and so on to the last, which contained 36 men, how many 
men were there in the whole body ? Ans. 160. 

9. The last term of a decreasing progression by differ¬ 
ence is 99, the number of terms is 30, and the common dif¬ 
ference is 3 ; what is the sum of the terms ? Ans. 1,665 


PROGRESSION BY QUOTIENT, 


OFTEN CALLED GEOMETRICAL PROGRESSION. 

Lesson 208. 

A row or series of numbers constantly increasing or de¬ 
creasing, so that if any number in it be divided by the one 
that precedes, the quotient will be the same in all parts of 
the row or series, is called a progression by quotient; thus, 
1,2, 4, 8, 16, is an increasing progression by quotient, 
and 125, 25, 5, 1, is a decreasing progression by quotient. 

The numbers which form a progression by quotient are 
called the terms. 

The quotient formed by dividing any term by the pre¬ 
ceding one is called the ratio. In a decreasing progres¬ 
sion by quotient, the ratio is less than 1. 

There are five things in progression by quotient, any 


What is called a progression by quotient ? Give an example of an in¬ 
creasing and of a decreasing progression by quotient ? 

What are called the terms ? What is called the ratio ? In a decreas¬ 
ing progression by quotient what is the ratio less than ? 

How many things are there in progression by quotient ? 




268 


PROGRESSION BY QUOTIENT. 


three of which being known, the other two may be found ; 
these things are, 

1 st, The first term, 

2d, The last term, 

3d, The number of terms, 

4th, The ratio, 

5th, The sum of all the terms. 

By examining these five things, we see that three differ¬ 
ent things may be selected in ten different ways. Now in 
each way three things are chosen, there are two to be 
found, so there are twenty different calculations to make, 
in order to obtain the answer to every kind of question that 
may arise in progression by quotient. 

We shall now point out the method of obtaining the an¬ 
swers to the most important questions. 

1 . A man commenced trade with $645, and by skilful 
management, doubled his capital every year for 6 years ; 
w r hat sum did he have at the end of that time ? 

OPERATION. 

2 $645 

2 64 


4 2580 

2 3870 


8 $ 4 1,2 8 0 Ans. 

2 

16 
2 

32 
2 

64 

2 . A man commenced trade with $2,025, but by unskil¬ 
ful management, possessed only § as much capital at the 


Explanation. We observe first 
that there are 7 terms, which are 
the sum he began with, and the 
sum he had at the end of each 
year. Now the most obvious way 
to obtain the answer is, to mul¬ 
tiply together $645 and 2, 6 times ; 
but this way of proceeding being 
tedious, we find the answer by 
multiplying 2 by itself 5 times, 
and then multiplying $645 by 
the result. See Multiplication, les¬ 
son 38. 


How many must be known to find the others? Name these things. 
What do we see by examining these five things ? 

Explain how example 1, lesson 208, is performed. 





PROGRESSION BY QUOTIENT. 


end of every year as he had at the 
he worth at the end of 4 years ? 

beginning ; 

what was 


OPERATION. 



2 numer. of f 

3 denom. of § 

$2025 


2 

3 

16 


4 

9 

12150 


2 

3 

2025 

$ 


8 

27 81 

)32400(400 

Ans. 

2 

3 

324 


16 

81 

00 



Explanation. We observe first, that there are 5 terms, 
which are the sum he began with, and the sum he had at 
the end of each year. Now the most obvious way to ob¬ 
tain the answer, is to multiply together $2,025 and f, 4 
times ; but this way of proceeding being tedious, we find 
the answer by multiplying § by itself 3 times, and then 
multiplying $2,025 by the result. See Multiplication, les¬ 
son 38. 

- Therefore, to find the last term, when the first term, the 
number of terms, and the ratio are known, 

Multiply the raliS by itself 2 times less than the number of 
terms, and multiply the result by the first term. 

3. What is the first term of a progression by quotient, 

the last term being 4, the number of terms 6" and the 
ratio £ ? Ans. 4,096. 

Explanation. Call the first term the last, and the last 
term the first ; the ratio then will be f or 4. 

4. Suppose the number of inhabitants in the United States, 
in 1820, was 10,000,000, and that the number doubles ev¬ 
ery 30 years, how many will there be in the year 2000, 
that is, after the number has doubled 6 times ? 

Ans. 640,000,000. 

5. If a man had but $1 at compound interest in the year 

1600, what would it have amounted to in 1828, allowing it 
to double every 12 years ? Ans. $524,288. 


Explain how example 2, lesson 208, is performed. 

How do we find the last term, when the first term, the number of 
terms, and the ratio are known ? 

23 * 






270 


PROGRESSION BY QUOTIENT. 


Lesson 209 . 

1. What is the sum of the following progression by quo¬ 
tient, 2, 8, 32, 128, the ratio of which we observe is 4 ? 

Explanation. We can obviously find the sum by adding 
the numbers together ; but this way of proceeding will be 
very tedious when there are many numbers. To find an 
easier method, multiply the terms, 2, 8, 32, all except the 
last, by the ratio, 4. 

The products are 8 32 128 

From these subtracting, 2 8 32 

we obtain 126, the difference between the first and last 
terms. This difference is plainly 3 times 2, 8, 32, since 
it is the difference between these numbers, and 4 times 
these numbers ; if we now divide 126 by 3, and add 128, 
the last term, to the quotient, we shall have the sum of 2, 
8, 32, 128. The operation is as follows : 

128 last term, 
subtract 2 first term. 

3)126 

42 

Add last term 128 

170 sum. Ans. 

2. What is the sum of the following progression by 
quotient, 486, 162, 54, 18, 6, 2, the ratio of which we see 
is £ ? 

Explanation. Direct addition being usually tedious, to 
find an easier method, multiply the terms 486, 162, 54, 18, 
6, all except the last, by the ratio 4, 

and from 486 162 54 18 6 

subtracting the products 162 54 18 6 2 

we obtain 484, the difference between the first and last 
terms. This difference is plainly § of 486, 162, 54, 18, 6, 
since it is the difference between these numbers and £ of 
them ; if we now divide 484 by f, and add 2, the last term, 


Explain how example 1, lesson 209, is performed. 



PROGRESSION BY QUOTIENT. 


271 


to the quotient, we shall have the sum of 486, 162, 54, 18, 
6, 2. The operation is as follows ; 

486 last term. 

Subtract 2 first term. 

484 

Divide by § 3 

2)1452 
726 

Add last term 2 

728 sum. Ans. 

Therefore, to find the sum of the terms, when the first 
term, the last term, and the ratio are known, 

Divide the difference between the first and last terms by the 
difference between l and the ratio , and add the last term to the 
quotient. 

3. A person has 2 parents, 4 grand-parents, 8 great- 

grand-parents, and so on ; now allowing 25 years to a 
generation, what number of ancestors had a person, born 
in 1825, descended from since the year 1325 ; supposing 
that the person’s ancestors had in no case married rela¬ 
tives ? Ans. 4,194,302. 

Explanation. Begin by finding the last term. 

4. A gentleman offered to sell a beautiful cottage to a 

young fop, and receive in payment 1 cent for the first 
window in it, 10 cts. for the second, 100 cts. for the third, 
and so on ; the young witling, thinking it a fine bargain, 
agreed accordingly ; what did the house cost him, there 
being 11 windows in it ? Ans. $111,111,111.11. 

5. What is the sum of the following progression by quo¬ 
tient, 4, 6, 9, &.c., the last term of which is 45f ? 

Ans. 128£. 

Lesson 210. 

1. What is the sum of the progression by quotient, 2, 1, 
L £> £> &c., continued on for ever ? Ans, 4. 


Explain how example 2, lesson 209, is performed. 

How do we find the sum of the terms, when the first term, the last 
term, and the ratio are known ? 



272 


PROGRESSION BY QUOTIENT. 


Explanation. After the progression has been continued 
on for a very great number of terms, the last term will be 
so near 0 in value, that it may be considered nothing. 

2. What is the sum of the progression by quotient, 1, 

£, ^ &.C., continued on for ever ? Ans. 1 

3. What is the sum of the progression by quotient, y 1 ^, 

tttW) &c., continued on for ever ? Ans. 

4. What is the value of the repeating decimals, .3333, 

&c. ? Ans. 

Explanation ... These decimals are equal to txtj Tinj-> 
nriHFj &-P- 

5. What is the value of the repeating decimals, 

.139139139, &c. Ans. £§f. 

Explanation. These decimals are equal to 

TTrtjirlnnrj &- c - 

6. If a farmer should sow 5 kernels of wheat, and its 

produce, every year for 9 years, how many bushels would 
there be in the last harvest, supposing that each harvest 
amounts to 10 times the quantity sowed, and that 8,000 
kernels make 1 pint ? Ans. 9,765 bu. 2£ pks. 

v 7. What is the amount of the following sums ; 384, 192, 
96, 48, 24, 12, 6, 3 ? Ans. 765. 

Note. We have given no rules to find the answers to many ques- ‘ 
tions in progression by quotient, but hardly one of those not noticed 
will occur in the course of a man’s life. However, when we find such 
a question, the answer can be found by an obvious though lengthy 
method. 


Lesson 211. 

PROGRESSION BY QUOTIENT APPLIED TO COMPOUND 
INTEREST. 

To obtain the amount of a sum at simple interest for 1 
year, say at 6 per cent., we multiply the principal by .06, 
and add the principal to the product, or multiply it by 1.06, 
which is obviously the same thing. 

To obtain the amount of a sum at compound interest for 


What have we given no rules for? Do such questions often occur? 
What if we find such a question ? 

In what two ways do we obtain the amount of a sum at simple inter¬ 
est for 1 year, say at 6 per cent. ? 




PROGRESSION BY QUOTIENT. 


273 

a number of years, say at 6 per cent., we can multiply it by 
1.06, the product by 1.06, and so on, multiplying by 1.06 
as many times as there are years. The original sum, 
then, and the amounts at the end of each year, form a pro¬ 
gression by quotient. The first term is the original sum, 
the number of terms is 1 more than the number of years, 
and the ratio is 1.06, 1.07, 1.08, &c., according to the per 
cent. 

1. What will $80 amount to in 12 years at 6 per cent., 

compound interest ? Ans. $160.98. 

Explanation. In order to perform this and the following 
examples with ease, examine the table in Compound In¬ 
terest. 

2. Find the amount of $6,525.12£ for 8 years, at 7 per 

cent., compound interest. Ans. $11,211.38. 

3. What is the compound interest on $1,000 for 10 

years, at 6 per cent. ? Ans. $790.85. 

4. What will $100 amount to in 9 years at 5 per cent., 

compound interest ? Ans. $155.13. 

5. How long will it take $50 to amount to $84.47 at 6 

per cent., compound interest ? Ans. 9 years. 

PROGRESSION BY QUOTIENT APPLIED TO DISCOUNT AT 
COMPOUND INTEREST. 

6. What is the present worth of $3,581.70 due in 10 

years, without interest, discounting at the rate of 6 per 
cent., compound interest ? Ans. $2,000. 

Explanation. The number of terms is 11, and the ratio 
evidently x .Vb 5 to find the last term. 

7. What principal at 6 per cent., compound interest, will 

amount to $143.45, in 8 years ? Ans. $90. 

8. What sum ought to be paid now to discharge a debt 
of $500, due in 6 years, without interest, supposing money 
to be worth 7 per cent., compound interest ? 

Ans. $333.17. 

9. What discount should be made for the immediate 
payment of $1,000, due in 5 years, without interest, money 
being worth 10 per cent., compound interest ? 

Ans. $379.08. 

Explanation. What is the present worth ? 

How can we obtain the amount of a sum at compound interest for 
any number of years, say at 6 per cent. P What do the original sum, 
and the amounts at the end of each year form? What is the first term ? 
Number of terms ? Ratio ? 



274 


ANNUITIES. 


ANNUITIES. 


Lesson 212. 


A sum of money payable at regular intervals during a 
certain time or for ever, is called an annuity ; thus, rents, 
pensions, salaries, &.C., are really annuities. 

If an annuity is not paid when due, it is said to be in 
arrears . 

When an annuity is not to commence until after a cer¬ 
tain time, it is said to be in reversion. 

The sum of the instalments of an annuity not paid, to¬ 
gether with the interest on each instalment after it becomes 
due, is called the amount. 


ANNUITIES AT SIMPLE INTEREST. 

1. The rent of a house was $100 a year, and the occu¬ 
pant paid nothing until 4 years after he entered it ; what 
was then due, 6 per cent, being the legal interest ? 

OPERATION. 

The rent of the fourth year being paid when due, is $100 


The amount of $100 for 1 year is. 106 

The amount of $100 for 2 years is. 112 

The amount of $100 for 3 years is. 118 


AN EASIER WAY. 

$118 amount of $100 for 3 years. 
100 rent of the fourth year. 

2)218 

109 

4 years. 

$436 Ans. 


Ans. $436 
Explanation. The 
operation by the first 
method is obvious ; by 
examining it, we see 
that the amounts of 
the rents form a pro¬ 
gression!^ difference; 
so we can get their 
sum in an easier way, 
by the rule for finding 
the sum of a progres¬ 
sion by difference. 


What is called an annuity ? What are really annuities ? 
When is an annuity said to be in arrears? In reversion ? 
What is called the amount? 

Explain how example 1, lesson 212, is performed. 


/ 






ANNUITIES. 


275 


Therefore, to find the amount of an annuity in arrears, 
at simple interest, 

Observe that the amounts of the several instalments, remain¬ 
ing unpaid, form a progression by difference, and find their 
sum, which is the whole amount, accordingly. 

2. A man had an annuity of $240 a year, during life, 

but the company it was purchased from becoming embar¬ 
rassed, the annuity remained unpaid, that is, in arrears, 
30 years ; what was then due, 6 per cent, being the legal 
interest ? Ans. $13,464. 

3. A man hired a house at a rent of $425 a year, but 

paid nothing until 10 years after the rent of the first year 
was due ; what did the rent then amount to, 7 per cent, 
being the legal interest ? Ans. $5,588.75. 

4. An annuity of $20 a month remained unpaid for 3 

years ; what was then due, 6 per cent, being the legal 
interest ? Ans. $783. 

5. What is the present worth of an annuity of $ 100 a 
year, to commence 1 year from this time, and to continue 
3 years, money being worth 6 per cent. ? 

operation. See Discount. 

$94.34 present worth of first instalment. 

89.286 present worth of second instalment. 

84.746 present worth of third instalment. 

$268.37 Ans. 

Therefore, to find the present worth of an annuity at sim¬ 
ple interest, 

Find the present worth of each instalment of the annuity, 
and the sum of these present worths will be the answer. 

6. What is the present worth of a salary of $800, to be 
paid at the end of every year from this time, and to con¬ 
tinue 4 years, 6 per cent, being the legal interest ? 

Ans. $2,792.13. 

7. What sum of ready money is equal to an annuity of 


How do we find the amount of an annuity in arrears, at simple in¬ 
terest ? 

Explain how example 5, lesson 212, is performed. 

How do we find the present worth of an annuity at simpld interest ? 




276 


ANNUITIES. 


$200, to commence 1 year from this time, and to continue 
5 years, money being worth 10 per cent. ? Ans. $778.52. 

Note. If an annuity is to continue a long time, the present worth, 
found in this way, will oflen be larger than a sum, the yearly interest 
of which is equal to the annuity ; thus, reckoning interest at 5 per cent., 
the present worth of an annuity of $100 a year to continue 50 years, 
will be more than $2,000, the yearly interest of which is $100. The 
rule then is absurd, though it may be used without much error to find 
the present worth of an annuity to continue for a short time. 


Lesson 213. 


ANNUITIES AT COMPOUND INTEREST. 

1. An annuity of $100 a year has not been paid for 4 
years ; what is the amount due on it, money being worth 
6 per cent., compound interest ? 

OPERATION. 

The amount of the fourth year, now due, is. .. .$100 


The amount of $100 for 1 year is. 106.00 

The amount of $100 for 2 years is. 112.36 

The amount of $100 for 3 years is. 119.1016 


Ans. $437.46. 


AN EASIER WAy. 

From $119.1016 last term, 
Subtract 100 first term. 


Diff S e the et r W aUo I ! -06) 19.1016 


318.36 

Add last term 119.10 


$437.46 Ans. 


Explanation. The 
operation by the first 
method is obvious ; 
by examining it, we 
see that the amounts 
of the instalments 
form a progression 
by quotient ; so we 
can get their sum an 
easier way, by the 
rule for finding the 
sum of a progression 
by quotient. 


Therefore, to find the amount of an annuity in arrears, 
at compound interest, 


If an annuity is to continue for a long time, what is said of its pres¬ 
ent worth ? Give an example. Is the rule correct then? When may 
it be used ? 

Explain how example 1, lesson 213, is performed. 










ANNUITIES. 


277 


Observe that the amounts of the several instalments, remain¬ 
ing unpaid, form a progression by quotient, and find their 
sum, which is the whole amount, accordingly . 

2. A man paid a certain sum to a company for an annu¬ 

ity of $250 a year during life, but the company becoming 
embarrassed, did not pay the annuity for 8 years ; what 
was the amount then in arrears, allowing 6 per cent., com¬ 
pound interest ? Ans. $2,474.37, 

Explanation. The table in Compound Interest shows 
the products of 1.05, 1.06, and 1.07 multiplied by them¬ 
selves any number of times up to 39. 

3. A man had a salary of $400 a year, payable quar¬ 

terly, but which was not paid for 2 years ; what was then 
due, allowing quarterly compound interest, at 7 per cent 
a year ? Ans. $850.75 

4. What is the amount of an annuity of $333 a year, 

which has been in arrears 5 years, allowing 5 per cent., 
compound interest ? Ans. $1,768.27. 

5. What is the present worth of a pension of $100 a 
year, the first instalment of which is to be paid in 1 year, 
and which is to continue 3 years, 6 per cent., compound 
interest, being allowed ? 


OPERATION. 

The present worth of first instalment is.$9 4.3 3 9 6 

The present worth of the last instalment is ... 8 3.9 6 2 

Divide by T ;$f the difference be- 1 0.3 7 7 6 

tween 1 and the ratio x 1.0 6 


622656 

103776 


.0 6)1 1.0 0 02 5 6 


1 8 3.3 3 7 
Add last term 8 3.9 6 2 


Ans. $2 6 7.3 0 


How do we find the amount of an annuity in arrears, at compound 
interest ? 


24 










278 


ANNUITIES. 


Explanation. To get the present worth of the first in¬ 
stalment, we multiply $100, by T ^ ; see Progression by 
Quotient, lesson 211 ; to get the present worth of the 
second instalment, we multiply $100 by the product of 
multiplied by itself, and so on. We see then, that the 
present worths of the instalments form a progression by 
quotient, the ratio of which is ; so we find the first 
and last terms, and then get the sum of the terms by the 
rule in Progression by Quotient. 

Therefore, to find the present worth of an annuity, dis¬ 
counting by compound interest, 

Find the present worth of the first and last instalments , 
and then find the sum of all the present worths, by the rule for 
finding the sum of all the terms in Progression by Quotient. 

6. What sum of ready money will purchase an annuity 

of $240 a year, to commence 1 year from this time, and to 
continue 9 years, money being worth 6 per cent., com¬ 
pound interest ? Ans. $1,632.41. 

7. What is the present worth of an annuity of $180 a 
year, to commence in 1 year, and to continue for ever, 
money being worth 7 per cent., compound interest ? 

Ans. $2,571.43. 

8. What is the present worth of an annuity of $500 a 
year, to commence in 6 years, and to continue 6 years, 
money being worth 7 per cent., compound interest ? 

Ans. $1,699.24. 

9. What annuity will $3,000 buy, to commence in 1 

year, and to continue 3 years, money being worth 6 per 
cent., compound interest ? Ans. $1,122.33. 

Explanation. What sum will purchase an annuity of 
$ 1 a year, to commence in 1 year, and to continue 3 years ? 
How many times is this sum contained in $3,000 ? 


Explain how example 5, lesson 213, is performed. 

How do we find the present worth of an annuity, discounting by 
compound interest ? 



PROMISCUOUS QUESTIONS. 


279 


PROMISCUOUS QUESTIONS 

IN 

PROGRESSION BY DIFFERENCE, &c. 

Lesson 214. 

1. An annuity of $200 a year remained unpaid 5 years ; 

how much did it then amount to, 7 per cent, being the legal 
interest ? Ans. $1,150.15. 

2. The poise of a steelyard, when placed in the first 

notch, balances 4 oz., and at every notch it is moved, it 
balances 2 oz. more than before ; in what notch is it when 
it balances 12 lbs. ? Ans. in the 95th. 

3. A countryman, in going to market, found the roads 

so bad that he went only 4 miles the first day ; however, 
as the roads grew better, he increased his speed, and com¬ 
pleted his journey in 6 days, going 14 miles the last day ; 
how much did he increase his speed each day, on an 
average ? Ans. 2 miles. 

4. By the preceding supposition, how far did the coun¬ 
tryman travel ? Ans. 54 miles. 

5. What sum of ready money will purchase an annuity 
of $300 a year, to begin in 1 year, and to continue for ever, 
money being worth 5 per cent., compound interest ? 

Ans. $6,000. 

6. There is a progression by quotient, the first term of 

which is 3, the number of terms 9, and ratio 4; what is the 
sum of all the terms ? Ans. 262,143. 

7. The first term of an increasing progression by differ¬ 

ence is 7, the common difference is 3, and the number of 
terms 12 ; what is the last term' ? Ans. 40. 

8. What sum of ready money is equal to 2 annual pay¬ 

ments of $500, the first in 1 year, money being worth 6 per 
cent., simple interest ? Ans. 918.13. 

9. A man had a salary of $700 a year, which was not 

paid for 3 years ; what was then due, allowing 6 per cent., 
compound interest ? Ans. $2,228.52. 

10. A merchant began trade with $3,000, and his gains 

were such that at the end of every year, he had §■ as much 
as he had at the beginning ; what was he worth at the end 
of 6 years ? Ans. $34,171.87 


280 


MONEY. 


MONEY. 

Lesson 215. 

Money is composed of gold and silver, metals which are 
extremely precious, because they are of great use in the 
arts. Copper, a valuable but much inferior metal, is also 
used for money. The value of gold and silver never arose 
from whim or fancy, but always has existed from the na¬ 
ture of things, because these metals are indispensable for 
the manufacture, preservation, or embellishment of many 
articles of necessity or convenience. You cannot eat or 
drink gold and silver, it is true, neither can you eat or 
drink an axe, a plough, or a steamboat, which are, never¬ 
theless, articles of great utility. 

Gold and silver, in the form of money, do not repre¬ 
sent property, but are property, and serve as a medium of 
exchange . They are admirably fitted for the purposes of 
exchange, because they are scarce and very precious, 
light in proportion to their value, clean, not liable to decay 
or be destroyed, &.c. If a man in Massachusetts has 500 
bushels of potatoes, and wishes to buy a farm in Michigan, 
he does not transport his potatoes to Michigan, and ex¬ 
change them for the farm, because potatoes are of small 
value there, would spoil in carrying, and the transportation 
would cost more labor than would be necessary to raise 
five times the quantity. On the contrary, he gets 2 or 300 
dollars for them, and carrying the money to Michigan, ex¬ 
changes it for land. 

Coins of gold, silver, or copper, are called true or real 
money. Money that has no coin to represent it, is called 
imaginary money, as a New England shilling, a mill, Sac. 
Accounts are kept in imaginary money as well as in real 
money. 


What is money composed of? What kind of metals are they ? Why 
are they precious ? What other metal is used for money ? Did the 
value of gold and silver arise from whim or fancy ? Why has it always 
existed ? What is said of eating and drinking gold and silver ? 

Do gold and silver, in the form of money, represent property ? What 
are they, and for what do they serve ? Why are they admirably fitted 
for the purposes of exchange ? What is a strong instance of the utility 
of money for the purposes of exchange ? 

What are called true or real money ? What is called imaginary 
money ? Give some examples ? In what money are accounts kept ? 



MOJNEY. 


% 281 

The following is a description of the money in which ac¬ 
counts are kept in the principal places where we trade. 

Continual alterations are taking place in the money of 
foreign countries, so that an accurate description will soon 
be erroneous. 

Note. The scholar should read the whole, and commit to memory 
the answers to the questions. 

Great Britain. 

Accounts in Great Britain are kept in pounds, shillings, 
pence, and farthings. 

4 farthings sign qr. make 1 penny, sign d. 

12 pence make 1 shilling, sign s. 

20 shillings make 1 pound, sign £ or /. 

Note. Accounts in this country w r ere formerly kept in pounds, shil¬ 
lings, pence, and farthings; 4 farthings made 1 penny, 12 pence 
1 shilling, and 20 shillings 1 pound, just as in the money of Great 
Britain ; the value of this money, however, was not the same as the 
English money, and varied in different parts of the country. In the 
origin this money had the same value as the English, but the govern¬ 
ments of the colonies having fabricated paper money, of no real 
value, it depreciated, so that a silver dollar would buy 6 shillings of Mas¬ 
sachusetts money, 8 shillings of New York money, &c. 


A dollar is estimated, 

In the money of Great Britain and Newfoundland, 
called English or sterling money, 

In the money of Canada and Nova Scotia, called 
Canada currency , 

In the old money of New England, Virginia, Ken¬ 
tucky, and Tennessee, called New England currency , 
In the old money of New York and North Caro¬ 
lina, called New York currency , 

In the old money of Pennsylvania, Delaware, Mary¬ 
land, and New Jersey, called Pennsylvania currency , 


* at 4s. 6d., or * 9 g 
I of a pound, 
at 5s., or | of a 
pound. 

| at 6s., or xo of a 
i pound. 

at 8s., or xq of a 
pound. 

at 7s. 6d., or § 
of a pound. 


What is said of alterations of money in foreign countries ? 

In what are accounts kept in Great Britain ? 

Recite the table of the money of Great Britain. 

In what were accounts formerly kept in this country? How many 
farthings made a penny, pence a shilling, and shillings a pound ? Was 
it of the same value as the English money? Was the value the same 
in all parts of the country ? What was the value in the origin ? Why 
did it alter P 

What is called sterling money ? Canada currency ? New England 
currency? New York currency ? Pennsylvania currency ? 

What is a dollar estimated at in sterling money? In Canada curren¬ 
cy ? In New England currency ? In New York currency ? In Penn¬ 
sylvania currency ? 

24* 



282 


MONEY. 


In the old money of Georgia and South Carolina, > at 45. 8 d., or 
called Georgia currency, ) of a pound. 


By 

this 
esti- * 
mate. 


£1 sterling ie worth. 

£1 Canada currency is worth. 

£1 New England currency is worth. 

£1 New York currency is worth .. 

£1 Pennsylvania currency is worth. 

£1 Georgia currency is worth. 


$4.44f 

4.00 

3.33J 

2.50 

2.66f 

4.234 


It is often convenient to know something of these old currencies, 
though they are no longer used in the United States. Federal money 
was established by Congress in 1786, and is now universally employed. 


Lesson 216. 

I 

France. 

10 centimes sign c. make 1 decime. 

10 decimes make 1 franc, sign fr. value $. 18 

Accounts have been kept in this money since 1795. 
The French do not mention their decime any more than 
we do our dime ; for instance, instead of 6 fr. 4 decimes, 
they say 6 fr. 40 c. 

Accounts were formerly kept in different money, thus ; 
12 deniers made 1 sou. 

20 sous made 1 livre, or livre Tournois, value $.18^,. 

Spain, Spanish America, and Spanish Colonies. 

Accounts are usually kept in reals, 

34 maravedies making 1 real. 

There are nine different kinds of reals, four of which are 
in general use ; these are, 

The real vellon , worth about 5 cents. 

The real of old plate , worth usually about 9 t ^j- cents 
The real of new plate, worth 10 cents. 

The real of Mexican plate, worth 12^- cents. 


What is called Georgia currency? 

What is a dollar estimated at in Georgia currency ? 

What is £1 sterling worth ? £1 Canada currency ? £1 New Eng¬ 
land currency ? £1 New York currency ? £1 Pennsylvania currency ? 
£1 Georgia currency ? 

What is said of these old currencies, and of Federal money ? 

Recite the table of the money of France. 

How long have accounts been kept in this money ? Do the French 
mention their decime ? What do they say for 6 francs 4 decimes ? 
What is the value of the real of Mexican plate ? 









MONEY. 


283 


In Spanish America, and the Spanish colonies, accounts 
are kept in reals of Mexican plate, t£nd hard dollars, worth 
$1. In Spain, when real is mentioned without explana¬ 
tion, real of old plate is understood ; 8 of these reals make 
what is usually called a dollar of exchange, an imaginary 
money, worth usually about 75 cents. This dollar is call¬ 
ed libra in Alicant and Valencia. In Gibraltar, accounts 
are kept in cob dollars , worth $1. The real of this place 
is worth 8^ cents 

Portugal, Brazil, and Portuguese Colonies. 

1,000 rees make 1 milree ; value in Portugal, $1.24 ; 
in Brazil, $1.05, and in Portuguese Colonies, $1.00. 

Russia. 

10 kopecs ~ make 1 grievener. 

10 grieveners make 1 ruble, value.$.75 

Business is done with paper money. The paper roiible 
is usually 60 or 80 per cent, below par. 

Prussia. 

12 pfennings make 1 good-groschen. 

24 good-groschen make 1 rix dollar, value.$.69 

This is the most usual way of keeping accounts. 


Sweden and Norway. 

12 runstycken make 1 skilling. 

48 skillings make 1 rix dollar, value.$1.07 


12 pfennings 
16 skillings 
6 marks 


Denmark. 
make 1 skilling, 
make 1 mark. 

make 1 rixbank dollar, value ... $.53 


Hamburg. 

12 pfennings make 1 skilling. 

16 skillings make 1 mark, value, about.$.30 

The word lubs, a contraction of Lubecks, is often put af¬ 
ter the money of Hamburg and Lubeck, to distinguish it 
from that of Denmark and other places ; thus we say, 5 
skillings lubs, 2 marks lubs. 


In what are accounts kept in Spanish America, and the Spanish 
colonies ? 









284 


MONEY. 


Certificates or evidences of money in the bank are trans¬ 
ferred and used as money. This bank money, called 
banco , is above par, and usually brings from 18 to 25 per 
cent, premium. A mark of bank money, or a mark banco, 
is usually worth about 35 cents. 

Belgium and Holland. 

100 cents make l florin or guilder, sign fl. value $.40 

Genoa and Leghorn. 

12 denari di pezza make 1 soldo di pezza. 

20 soldi di pezza make 1 pezza, value, about $.90 

also , 

12 denari di lira make 1 soldo di lira. 

20 soldi di lira make 1 lira, value, about..... $.16 

Rome. 

10 baiocchi make 1 paolo. 

10 paoli make 1 scudo or Roman crown, value $1.00 
Naples. 

10 grani make 1 carlino. 

10 carlini make 1 ducato, value. $.80 

X 

Sicily. 

20 grani make 1 taro. 

12 tari make 1 scudo or Sicilian crown, value $.95 


30 tari make 1 oncia, value...2.40 

Malta. 

20 grani make 1 taro. 

12 tari make 1 scudo, value.. $.40 

30 tari make 1 pezza, value.1.00 


Venice. 

10 millesimi make 1 centesimo. 

10 centesimi make 1 lira di Austria, value... $.16 
Accounts have formerly been kept here in many different 
ways. 

Vienna and Trieste. 

4 pfennings make 1 kreuzer. 

60 kreuzers make 1 florin or guilder, value... $.48 
1£ florin make 1 rix dollar of account. 







MONEY. 2§5 

Turkey, Greece, and the northern states of Africa. 

The only invariable currency in these countries is com¬ 
posed of Spanish hard dollars and parts. In Turkey and 
Egypt money is usually reckoned by the asper and para , 
3 aspers making 1 para, and 

40 paras making 1 piastre, or Turkish dollar, value 
from 5 to 50 cents. 

EAST INDIES. 

Mauritius, or the Isle of France. 

20 sous make 1 livre. 

10 livres make 1 dollar, value. $1.00 

Also, 100 cents make 1 dollar. 

Hindostan. 

Before European colonies were established in Hindos¬ 
tan, the principal currency was the sicca rupee, a silver 
coin, worth about 58 cents. This coin was also used as a 
weight. The British possessions in Hindostan are now 
divided into three presidencies, Bengal, Bombay, and 
Madras, the money of which differ. 

Calcutta, in Bengal. 

12 pice make 1 anna. 

16 annas make 1 rupee, value, about.$.46 

A lac of rupees is 100,000 rupees, and a crore of rupee* 
is 100 lacs, or 10,000,000 rupees. 

Bombay. 

100 rees make 1 quarter. 

4 quarters make 1 rupee, value, about.. $.45 

Madras. 

Money is reckoned in rupees, worth about $.45 ; the 
rupee is divided into halves, quarters, eighths and six¬ 
teenths. The sixteenth is called the anna. 

This is the new method, adopted in 1818, and is the only 
intelligible way of reckoning money in the place. 

Bencoolen, in Sumatra. 

8 satellers make 1 soocoo. 

4 soocoos make 1 dollar, value,. 


$1.15 





MONEY. 


m 


Batavia, in Java, 
make 1 stiver, 
make 1 dubbel. 
make 1 skilling. 

make 1 florin or guilder, value,.. .$.40. 
Canton, in China. 


10 cash make I candarine. 

10 candarines make 1 mace. 

10 mace make 1 tale, value,.$1.48. 

Japan. 

10 candarines make 1 mace. 

10 mace make 1 tale, value,.. $1.43. 


Bargains can usually be made in Spanish dollars in any 
part of the East Indies. 

WEST INDIES. 

British Islands. 

Accounts are kept in pounds, shillings, pence and far¬ 
things, in the British West India islands, in the Swedish 
islands of St. Bartholomew, and by the British settlers on 
the French islands. 

A pound in Jamaica and Bermuda currency is worth $3.00 

A pound inBarbadoes currency is worth.3.20 

A pound in Bahama currency is worth.4.28f 

A pound in the currency of the Leeward and Wind¬ 
ward Islands, employed also on St. Bartholomew, 
and by the British settlers on the French islands, 
is worth.2.22§ 

French Islands. 

The French, in these islands, reckon money as follows ; 
12 deniers make 1 sou. 

20 sous make 1 livre, value.. . $.11£ 

St. Domingo, or Hayti. 

Accounts are kept in dollars and cents, as in the United 
States, but the dollar is worth only..$.66 

Danish Islands. St. Thomas, St. John, and Santa Cruz. 

it is cusiomary to keep accounts in rix dollars and 
cents, but the rix dollar, of colonial currency, is worth 
only...$.64 


5 doits 

2 stivers 

3 dubbels 

4 skillings 










COINS. 


287 

Dutch Settlements. St. Eustatia, St. Martin, and Cur- 
azoa. 

6 stivers make 1 real or skilling. 

8 reals make 1 piastre, value about.$.73 

Surinam, Berbice, Demarara, and Esseq,uibo. 

8 doits make 1 stiver. 

20 stivers make 1 guilder, value about.$.33£ 

Spanish Islands. See Spain. 


COINS. 

Lesson 217. 

Civilized nations coin the precious metals in pieces of 
certain values, so that they can be employed in making ex¬ 
changes without weighing them, or examining their purity. 
Uncivilized or half civilized people, like many in the East 
Indies, often use these metals in bullion as money, and 
every time each ingot or piece is exchanged, are obliged to 
weigh it, and estimate its purity. 

Gold and silver, when pure, are exceedingly soft and 
ductile ; so if they were coined in this state they would 
soon be defaced and worn out. Therefore, they are mixed 
or alloyed with a very small quantity of copper, which 
makes them quite hard, and but little impairs their duc¬ 
tility, incorruptibility, and other useful properties. There 
are two things then on which the value of each coin de¬ 
pends ^ its weight, and the quantity of pure gold or silver 
which it contains. 

We have mentioned the coins of the United States, and 
their values, in lesson 90, Federal Money ; it is proper to 


Why do civilized nations coin the precious met&ls ? How do uncivil¬ 
ized or half civilized people use these metals as money ? What are they 
obliged to do every time each ingot or piece is exchanged ? 

What is the nature of gold and silver when pure ? What if they 
were coined in this state? What, therefore, is done to them ? What 
effect does this have ? On what two things does the value of each coin 
depend ? 






288 


corns. 


observe further, that the gold pieces, coined since 1834, 
contain of alloy ; the eagle containing 232 grains of 
pure gold, and 26 grains of alloy, or 258 grains of stand¬ 
ard gold. The alloy consists of copper and silver, the 
copper never being less than half of it. 

The silver pieces contain of alloy ; the dollar hav¬ 
ing 371^ grains of pure silver, and 44£ grains of pure cop¬ 
per, or 416 grains of standard silver. The copper coins 
consist of pure copper. 

The gold pieces coined before 1834, contained about T \g 
of alloy ; the eagle consisting of 247£ grains of pure gold, 
and 22|- grains of alloy, or of 270 grains of the standard 
gold of that time. These eagles are now worth $10.66£, 
and the half and quarter eagles in proportion. 

The reason for altering the size of the eagle was this ; 
when the first eagles were coined, y 1 ^ of an ounce of gold 
was worth as much as 1 ounce of silver, so they put as 
many grains of gold in ^ of an eagle, as they put grains 
of silver in a dollar. In 1834, of an ounce of gold had 
become worth as much as 1 ounce of silver, so they put 
about ^ as many grains of gold in y 1 ^ of an eagle as there 
are grains of silver in a dollar. 

The gold coins of Great Britain, France, Spain, Mexi¬ 
co, Columbia, Portugal, and Brazil, were made current in 
1834, by law of Congress, and their values fixed as fol¬ 
lows ; 

Gold coins of Great Britain, Portugal and 

Brazil, at least 22 carats fine, at . ...$.948 a pwt. 

Gold coins of France, -$j fine, at.931 a pwt. 

Gold coins of Spain, Mexico, and Colombia, 

20 carats 3 T 7 ^ grains fine, at.899 a pwt. 

Foreign gold coins usually sell at a premium. The fol¬ 
lowing are some of the principal foreign gold coins, with 
what may be considered their par value in this country ; 


Value. 


f Guinea, 21 shillings, £ guinea in 



^ Seven-shilling piece, 


$5.07 

4.85 

1.70 


What is the value of the eagles coined before 1834 ? 

Why was the size of the eagle altered ? 

What gold coins were made current by law of Congress ? 









COINS, 


289 


France. < 


Spain, 

Spanish 

Colonies, 

. Mexico, and 
Colombia. 


(* Napoleon, 20 francs, Double Napole 

on in proportion, .. 

[New Louis,. 

'Doubloon 
vellon, 

Doubloon 
I vellon, 

Doubloon or pistole, 80 reals vellon, 

Escudo, 40 reals vellon,. 

Coronilla, 20 reals vellon,. 

Patriot doubloon, parts in proportion. 


of 8 escudos, 320 reals 
of 4 escudos, 160 reals 


Portugal 

and 

Brazil. 


'Dobra, 12,800 rees,. 

Joanese, 6,400 rees, .... 
Half joanese, 3,200 rees, 

Escudo, 1,600 rees,. 

Half escudo, 800 rees, .. 


Russia, Half Imperial, or 5 ruble piece,.... 

Pncssia, Ducat,... 

Sweden and Norway, Ducat,. 

Denmark, Ducat-current, coined since 1757 

Hamburg, Ducat,. 

Holland and Belgium, 10 guilder piece,. 

Genoa and Leghorn, Sequin,.. 

Rome, Sequin, .. 

Naples and Sicily, Oncia,..... 

Malta, Spanish gold coins. 

Venice, Sequin,... 

Vienna and Trieste, Ducat,.. 

Turkey, Sfc., Sequin, coined since 1818, .... 

C Star pagoda,. 

East ladies, < Gold rupee,.* ..., 

( Japanese new copang,. 

West Indies, Spanish gold coins. 


$3.85 

3.85 


16.00 

8.00 

4.00 

2.00 

1.00 

15.53 

17.30 

8.65 

4.32 

2.16 

1.08 

3.92 

2.27 
2.23 
1.80 

2.28 
4.00 
2.31 
2.25 
2.50 

2.31 

2.29 

1.83 

1.80 

7.10 

4.92 


SILVER COINS. 

The dollars of Mexico, Peru, Chili, and Central America, 
weighing not less than 415 grains each, and at least 10 oz. 
15 pwts. fine, with those re-stamped in Brazil, of like 
weight and fineness, were made current in 1834, by law of 


What foreign silver coins were made current by law of Congress ? 


25 
































290 


COINS. 


Congress ; the five franc piece of France, weighing not 
less than 384 grains, and at least 10 oz. 16 pwts. fine, was 
made current by the same law, and the value fixed at 93 
cents. 

The following are some of the principal foreign silver 
coins, with what may be considered their par value in this 
country; 

Value. 

( Crown, 5 shillings, ... $1.10 

* Crown, 2s. 6d.,.55 

Shilling, ..22 

Sixpence,.11 

5 franc piece, .93 

2 franc piece,.37^ 

Franc, .. 18^ 

Piece of 50 centimes, ..09 T ^y 

Piece of 25 centimes,.04*g 


coined 

Great Britain , since 
1816, 


France . 


Spain, 

Spanish Colonies, 
and 

Spanish America. 


Dollar,.. 

. 1.00 

* dollar. 


* of a dollar,... 

.25 

* of a dollar,. 

.12* 

jJg- of a dollar, . 

.06* 


These are the same pieces that cir¬ 
culate in our country. Spanish dol¬ 
lars are to be found in all parts of the 
world, and are more extensively used 
than any other money. 

Portugal and ( New crusado, 480 rees,. $.60 

Brazil, ( Teston, 100 rees,.12* 

Russia, Ruble,.75 

Prussia, Rix dollar,. 69 

Sweden and Norway, Rix dollar,. 1.07 

Denmark, Rixbank dollar,. .53 

Hamburg, Crown dollar,. 1.09 

Holland and Belgium, Florin or guilder,.40 

Leghorn, Francescone,. 1.05 

Rome, Scudo or Roman Crown,. 1.00 

Naples and Sicily, Ducat,.80 

Malta, Spanish dollar,. 1.00 

Venice, Ducat-effective,.77 

Vienna and Trieste, Rixdollar, 2 guilders,.96 

Turkey, Greece, and the Northern States of Africa , Span¬ 
ish dollars. 






























EXCHANGE. 


291 


C Calcutta, sicca rupee,. . $.48 

East Indies , < Bombay and Madras, silver rupee, .45 
( Japanese schuit,. 5.86 


Rest of East Indies with West Indies, Spanish dollars. 


EXCHANGE. 

Lesson 218 . 

1. What sum in Canada currency, must you receive for 

$87.25 ? Ans. £21 16s. 3d. 

Explanation. How many pounds and decimals of a 
pound are there in this sum ? Then how many pounds, 
shillings, &c., are there ? 

2. What is the value of £25 12s. 8d. 2 qrs. sterling in 

Federal money ? Ans. $113.94. 

Explanation. Change to pounds and decimals of a 
pound first ? 

3. What is £1,000 in Newfoundland currency worth ? 

Ans. $4,444.44. 

4. What is the value of $817.15 in sterling or English 

money ? Ans. £183 17s. 2^d. 

5. Change the following sums in sterling to Federal 

Money; £ 101 4s. 6d., £98 15s. 6d., £250 8s. 3d., and £49 
11s. 9d. Ans. $2,222.22. 

6. A man bought furs in Montreal to the amount of £78 

11s. 8d. ; how many dollars and cents must he give in 
payment ? Ans. $314.33^. 

7. The salary of the governor of Massachusetts is 

£1,100 New England currency; what is the amount in 
Federal money ? Ans. $3,666.67. 

8. Change $637.50 to New England currency ? 

Ans. £191 5s 


Lesson 219. 


1. How many dollars and cents are there in £750 18s. 
Georgia currency ? Ans. $3,218.14. 





292 EXCHANGE. 

2. Change $667.37 to Georgia currency. 

Ans. £155 14s. 4d. 3qrs. 

3. How many dollars and cents are there in £156 8s. 

9d. New York currency ? Ans. $391.09. 

4. Change $800 to New York currency. Ans. £320. 

5. Suppose a man in Delaware should recover £9,000' 

on a suit for money due his ancestors in 1780 ; what sum 
in Federal money should he receive ? Ans. $24,000. 

6. Change $1,700 to Pennsylvania currency. 

Ans. £637 10s 

7. How many pounds sterling are there in £200 New 

England currency ? Ans. £150. 

8. A man in London owes me £100 5s. 4d., and I am 

indebted to him $500 ; what sum in English money must I 
pay to balance the account ? Ans. £12 4s. 8d. 

9. Multiply £15 5s. 4d. sterling by 5, and change the 

product to Federal money. Ans. $339.26. 

•x 

Lesson 220. 

1. If you owe a merchant in St. Petersburg, Russia, 
12,000 paper rubles, worth 25 cents a piece, and he re¬ 
quests payment in Spanish dollars, allowing you 7 per 
cent, on them, what number of dollars must you send ? 

Ans. $2,803^. 

2. How many half joaneses, called half joes, must be 

shipped from Rio Janeiro, Brazil, to New York, to pay a 
debt of $6,897.75, Portuguese gold being 5 per cent, above 
par in New York ? Ans. 1,520 t 6 ^j 8 0 . 

3. If a merchant in Tampico, Mexico, owes you 30,000 

reals, what number of patriot doubloons should you receive 
in payment, these doubloons being 3 per cent, above par, 
and worth $16 apiece ? Ans. 234 T y o %. 

4. How many dollars in specie must be sfent to Havre, 

France, to pay a debt of 7,000 francs, specie in France 
being 2 per cent, above par ? Ans. $1,276 1 % 7 5 . 

5. A merchant in Liverpool owes a mercantile house in 

Boston $10,000 ; how many sovereigns must be shipped to 
Boston to pay the debt, English gold being 1 per cent, 
above par ? Ans. 2,041 

6. A merchant sold a quantity of flour in Paramaribo, 

Surinam, for 15,000 guilders ; how many Spanish dollars 
should he receive in payment ? Ans. 5,000 


EXCHANGE. 


293 


7. If you sell a cargo of rice in Amsterdam, Holland, 

for 24,000 florins, take your pay in 10 guilder pieces at 
par, and sell them at the United States mint for $4.03 a 
piece, what sum in Federal money do you obtain for the 
cargo ? Ans. $9,672., 

8. A merchant bought some tea in Canton for 800 Chi¬ 

nese tales, and paid for it in American coin at par ; what 
sum did he pay ? Ans. $1,184. 

9. A young American in Palermo, Sicily, bought some 

articles for 85 scudi 9 tari ; what sum in Federal money 
will pay for the articles ? Ans. $81.46. 

10. If you sell 90 quintals of cod fish in Guadaloupe, 
one of the French West India islands, for 3,600 livres, 
how many Spanish dollars should you receive in payment ? 

Ans. 400. 


Lesson 221. 

1. If you buy a quantity of saltpetre in Calcutta, for 

2,300 rupees, what number of Portuguese half joes, at 4 
per cent, above par, and what odd change in dollars and 
cents, will pay for it ? Ans. 235 half joes and $2.19. 

2. A merchant bought silks in Genoa to the amount of 

43,264 lire ; how many Spanish doubloons, at 4 per cent, 
above par, will pay for the silks ? Ans. 416. 

3. If a merchant in Kingston, Jamaica, owes you £645 

6s. 8d. currency, what number of Spanish dollars must he 
send you to cancel the debt ? Ans. 1,936. 

4. A trader sent a cargo of lumber to Barbadoes, and 
sold it for ,£600 currency ; the whole cost of the lumber 
delivered being $925,374, what profit did he make upon it ? 

Ans. $994.62^. 

5. What sum in Federal money is equal to £964 8s. 4d. 
of the currency of the Leeward Islands ? Ans. $2,143.15. 

6. What sum in Federal money must be given for a 

quantity of salt valued at £1,000 currency in Turks Isl¬ 
and, one of the Bahamas ? Ans. $4,285.71. 

7. What sum in Federal money do 108 florins 36 kreuz-? 

ers of the money of Trieste make ? Ans. $52.13. 

8. Change 8,736.45 dollars, in the currency of the 
Danish West India islands, to Federal money. 

Ans. $5,591.33. 

9. Change 1,988 marks banco 15 skillings of Hamburg, 

25* 


294 


EXCHANGE. 


to Federal money, estimating the mark banco at 34 cents ? 

Ans. $676.24. 


Lesson 222. 

It is not customary for merchants engaged in foreign 
trade, to pay in specie for the goods they import, or to 
receive specie for the goods they send abroad. The busi¬ 
ness of exchange is done by an easier and less risky 
method. For instance, John Smith, of New York, owes 
Peter Brinkerhoff, of Amsterdam, Holland, 6,000 florins ; 
now, if Smith has a creditor in Amsterdam who owes him 
enough to cancel the debt, he sends him an order to pay 
Brinkerhoff; if he has no such creditor, he goes to Charles 
Brown, who has, say 10,000 florins due him from Thomas 
Van Horne, of Amsterdam, and gets an order from Brown 
directing Van Horne to pay Brinkerhoff the 6,000 florins. 
There is evidently no risk in transmitting this order to 
Amsterdam. 

Such an order is called a bill of exchange, and is also 
styled bill or exchange on Amsterdam, or simply New York 
on Amsterdam. Exchanges between distant cities, as Bos¬ 
ton and New Orleans, are made in the same way, but the 
order is called a draft. 

In the preceding instance, if Smith obtains the order 
or bill on Amsterdam for 6,000 florins by paying the 
amount of that sum to Brown, exchange on Amsterdam 
is said to be at par ; if he pays more than the amount 
of 6,000 florins for the accommodation, exchange on Am¬ 
sterdam is said to be above par, in which case exchange 
is unfavorable to this country ; if he gets the bill for less 
than the amount of 6,000 florins, exchange on Amsterdam 
is said to be below par, in which case exchange is favor¬ 
able to this country. 

Is it customary for merchants engaged in foreign trade, to make their 
exchanges by means of specie ? Give an example of an easier and less 
risky method. Is there any risk in transmitting this order to Amster¬ 
dam ? 

What is such an order called ? What is it also styled ? How are ex¬ 
changes between distant cities, as Boston and New Orleans, made, and 
what is the order called ? 

In the preceding instance, when is exchange on Amsterdam at par? 
^bove par ? Below par ? 

In which qase is the exchange favorable to this country ? Unfavor¬ 
able ? 



EXCHANGE. 


295 

There is a continual variation in the rates of exchange 
between different countries, so that they may be favorable 
to a place one time, and presently unfavorable to it. 

1. Exchange on Cadiz, Spain, being worth 68 cents a 

dollar of exchange, what must I give for a bill of exchange 
on Cadiz, for 1,967 dollars of exchange, 7 reals and L5 
maravedies ? Ans. $1,338.19. 

2. A merchant in Havre, France, gave 10,400 francs for 
a bill of exchange on New York for $1,876.37 ; was ex¬ 
change in that place on New York above or below par, 
and what per cent. ? Ans. above par about 3 per cent. 

3. What sum in Federal money will a bill of exchange 

on Liverpool for £4,000 cost, exchange on that place be¬ 
ing 8 per cent, above par ? Ans. $19,200. 

4. How many pounds, shillings, &c., must a merchant in 

Jamaica give for a bill of exchange on Baltimore for 
$1,600, exchange on Baltimore being 3 per cent, above 
par ? Ans. £549 6s. 8d. 

5 What must I give for a bill of exchange on Calcutta 
for 1,835 rupees, 12 annas, 3 pice, at the rate of 50 cents a 
rupee ? Ans. $917.89. 

6. How many rubles, grieveners, &c., must a merchant 

of St. Petersburg, Russia, give for a bill of exchange on 
New York for $8,232, exchange on New York being 4 per 
cent, above par ? Ans. 11,415 rubles 4 kopecs. 

7. What must a merchant of Philadelphia give for a 

draft or bill of exchange on Norfolk for $2,000, at 2 per 
cent, above par ? Ans. $2,040. 


Lesson 223. 

1. A merchant in Boston owes 8,000 paper rubles in St. 
Petersburg, Russia ; being unable to purchase a bill of 
exchange on that place, he obtains one on London, and 
sending it to his correspondent' there, directs him to pur¬ 
chase a bill of exchange for 8,000 rubles on St. Peters¬ 
burg, and remit it to the creditor ; what sum in Federal 
money will pay the debt, if 1 dollar will purchase 4s. 2d. 


What is said of variation in the rates of exchange ? 



296 


EXCHANGE. 


exchange on London, and £1 in London will purchase 12 
rubles exchange on St. Petersburg ? Ans. $3,200. 

Explanation. See Chain Rule. 

2. A merchant in New York has a creditor in Hamburg, 

to whom he owes 9,000 marks banco ; this merchant, hav¬ 
ing correspondents in London, Havre in France, and Am¬ 
sterdam in Holland, draws a bill of exchange on London, 
for a proper amount, and directs London to draw on 
Havre, Havre on Amsterdam, and Amsterdam on Ham¬ 
burg ; what sum in Federal money will discharge the 
debt, if exchange of New York on London is 1 dollar for 
4s. 2d., exchange of London on Havre 1<£ for 24 francs, 
exchange of Havre on Amsterdam 2 francs for 1 florin, 
and exchange of Amsterdam on Hamburg 3 florins for 4 
marks banco ? Ans. $2,700. 

3. What is the gain or loss by remitting this way, instead 
of sending directly to Hamburg, exchange of New York 
on Hamburg being 4 per cent, above par ? 

Ans. the gain is $576. 

4. A merchant in Palermo, Sicily, owes 7,500 rix dol¬ 
lars in Stockholm, Sweden ; he is unable to purchase a 
bill of exchange on Stockholm, but can purchase one on 
London or on Havre, in either of which places his corre¬ 
spondents can purchase one on Stockholm, and send it to 
his creditor. Now, had he better remit the money by way 
of London, exchange of Palermo on London being 5 scudi 
for <£1, and exchange of London on Stockholm 4s. 2d. for 
1 rix dollar, or by way of Havre, exchange of Palermo on 
Havre being 1 scudo for 5 francs, and exchange of Havre 
on Stockholm 5 francs 25 centimes for 1 rix dollar ? 

Ans. he will save 62£ scudi in remitting by way of Lon¬ 
don. 

5. Suppose a merchant in Boston owes $5,000 in 

Charleston, South Carolina, between which places ex¬ 
change is at par, and remits a proper amount by means of 
drafts and correspondents from Boston to Philadelphia, 
from Philadelphia to Baltimore, and from Baltimore to 
Charleston ; how much will he save by this circuitous 
remittance, if exchange between Boston and Philadelphia 
is 2 per cent, in favor of Boston, between Philadelphia and 
Baltimore 1 per cent, in favor of Philadelphia, and between 
Baltimore and Charleston at par ? Ans. $149. 


WEIGHTS AND MEASURES. 


297 


Note. When money is remitted in a circuitous way, we must pay 
commission to the various correspondents or agents who transact part 
of the business; there is usually more interest lost than in a direct re¬ 
mittance ; and besides, there is considerable risk attending several 
operations transacted at a distance from us. 


WEIGHTS AND MEASURES. 

Lesson 224. 

The following is an account of the most important 
weights and measures in the principal places where we 
trade. Their value is expressed in the weights and meas¬ 
ures of the United States ; the gallons in wine measure, 
and the pounds in avoirdupois weight. 

Note. The scholar should read the whole, and commit to memory 
the answers to the questions. 

Great Britain. 

The weights and measures of Great Britain are the 
same as those of the United States, described in Compound 
Numbers, except the imperial gallon. See lesson 95, Com¬ 
pound Numbers. 

France. 

The old weights and measures of France are as fol¬ 
lows ; 


Foot, 

12.7893 

inches. 

Aune, or ell, 

46.85 

inches. 

Toise, 

6 

feet. 

Mile, £ of a lieue or league, 

1.212 

mile. 

Boisseau, 

.369 

bushel. 

Quart, of a muid, 

1.967 

quart. 

Commercial pound, or livre, 

1.08 

pound. 


New Weights and Measures. 

The French, during their revolution, ascertained the 
length of a quarter of a meridian of the earth, by means of 


What is said of remitting money in a circuitous way ? 

What are the weights and measures of Great Britain the same as? 
What did the French ascertain during their revolution ? 




298 


WEIGHTS AND MEASURES. 


some very exact observations and measures. A quarter 
of a meridian is a line running north and south, beginning 
at the equator, and ending at the pole. T<n> oWcny P ar * 
a quarter of the meridian was called a metre , a b rench 
word for measure. All measures, of whatever kind, were 
derived from the metre, and arranged in decimal propor¬ 
tions, like our dollars, dimes, cents and mills. They 
placed before any measure the Latin terms 
deci to express T V of it, 
centi to express of it, 
milli to express of it, 

And the Greek terms 

deca to express 10 times the number, 
heclo to express 100 times the number, 

kilo to express 1,000 times the number, 
myria to express 10,000 times the number. 


10 milli-metres 
10 centi-metres 
10 deci-metres 
10 metres 
10 deca-metres 
10 hecto-metres 
10 kilo-metres 


Thus, in Long Measure, 


make 

1 

centi-metre, 

English feet, 
value .032809167 

make 

1 

deci-metre, 

.32809167 

make 

1 

metre, 

3.2809167 

make 

1 

deca-metre, 

32.80,9167 

make 

1 

hecto-metre, 

328.09167 

make 

1 

kilo-metre, 

3,280.9167 

make 

1 

myna-metre. 

, 32,809.167 


Land or Square Measure. 

The unit of square measure is the square of the de¬ 
cametre, called the are. value about 1.076.4414 square feet. 


Dry and Liquid Measure. 

The unit of these measures is the cube of the decimetre, 
called the litre, value about .0353171 cubic feet. 


What is a quarter of a meridian ? What was called a metre ? How 
were all measures derived and arranged ? What did they place before 
any measure ? 

Recite the table for long measure, and state the length of the metre 
to hundredths. 

What is the unit of land or square measure ? What is it called ? 

What is the unit of dry and liquid measures ? What is it called ? 



WEIGHTS AND MEASURES. 


299 


Cubic or Solid Measure. 

The unit of cubic measure is the cube of the metre, 
called the stere, value about 35.3171 cubic feet. 

Weight. 

The unit of weight is the weight of a* cubic centi-metre 
oi distilled water, called the gramme , value about 15.434 
grains. 

The new system of weights and measures is not yet 
employed universally, except among theiearned. 

Spain, Spanish America, and Spanish Colonies. 


Foot,.11.128 inches. 

Vara, or yard,.33.384 inches. 

Common league,.4.291 miles. 

Fanega, y 1 ^ of a cahiz,. 1.599 bushel. 

Arroba of wine,. 4.245 gallons. 


Pound, of an arroba of weight,... 1.0144 pound. 


Portugal, Brazil, and Portuguese Colonies. 


Foot,. 

Vara, or yard,. 

Mile,. 

Fanga, ^ of a moyo, . 
Almuide of Lisbon,.... 
Pound, of an arroba, 


12.944 inches. 
43.2 inches. 
1.25 mile. 

1.535 bushel. 
4.37 gallons. 
1.0119 pound. 


Russia. 

Foot,.. 

Arsheen, or yard,. 

Werst, or Russian mile,.. 

Chetwert,. 

Wedro,..... 

Pound, of a pood,. 


..13.75 inches. 
..28. inches. 
3500. feet. 

.. .5.952 bushels. 
.. .3.246 gallons. 
... .9026 pound. 


What is the unit of cubic measure ? What is it called ? 

What is the unit of weight ? What is it called ? 

What should you call jq of an are ? 100 ares? 1,000 ares ? 155 of a 
litre? 10 litres ?'10,000 litres ? 15 of a stere ? 1,000 steres? 10,000 
steres ? of a gramme ? 10 grammes ? 100 grammes? 

Is the new system of weights and measures universally employed ? 




















300 


WEIGHTS AM) MEASURES. 


Prussia. 

Foot, or Rhineland foot,. 

Ell, or yard,.. 

Mile,. 

Sheffel,. 

Eimer,.... 

Commercial pound,. 


12.356 inches. 
26.256 inches. 
4.684 miles. 
1.5594 bushel. 
18.14 gallons. 
1.0311 pound. 


Sweden and Norway. 


Foot,.11.684 inches. 

Ell, or yard,.23.368 inches.’' 

Mile,. 6.648 miles. 

Kann,.297 peck. 

Kann,.69 gallon. 

Pound, victualie weight,.9376 pound. 

Denmark. 

Foot, or Rhineland foot,.12.356 inches 

Ell, or yard,.24.712 inches. 

Mile,. 4.684 miles. 

Barrel, or toende,. 3.9472 bushels. 

Viertel, ^ of a hogshead,.2.041 gallons. 


Commercial pound, of a centner, ... 1.103 pound. 


Hamburg. 

Foot,.. 

Hamburg ell, or yard,. 

Mile,. 

Fass,. 

Viertel, ^ of an ahm,. 

Commercial pound,. 


11.289 inches. 
22.578 inches. 
4.684 miles. 
1.494 bushel. 
1.912 gallon. 
1.068 pound. 


Belgium and Holland. 


Amsterdam foot,. 

Amsterdam ell, or yard, 

Schepel,. 

Stoop, . 

Aam,.. 

Commercial pound, . ... 


11.147 inches 
27.0797 inches. 

. .7892 bushel. 

, 5.125 pints. 
.41. gallons 
. 1.0893 pound. 































WEIGHTS AND MEASURES. 


301 


Genoa. 

Palmo,. 9.725 inches. 

Braccio, or yard,.22.692 inches. 

Mina,. 3.426 bushels. 

Barile of wine, £ of a mezzorala,.19.61 gallons. 

Commercial pound,.76875 pound 

Leghorn. 

Braccio, or yard, £ of a canna, . . x .23.25 inches. 

Sacco,.2.06 bushels. 

Barile of wine,.12. gallons. 

Pound,.75 pound. 

Rome. 

Foot,.11.72 inches. 

Mile,. 925 mile. 

Rubbio,. 8.356 bushels. 

Barile of wine,. 15.409 gallons. 

Pound,.7477 pound. 

Naples. 

Palmo,.10.375 inches. 

Canna,.83. inches. 

Tomolo, -g^ of a carro,. 1.451 bushel. 

Barile of wine,.11. gallons. 

Rottolo, yiitt of a gross cantaro, .. 1.965 pound. 

Pound, of a neat cantaro,.7067 pound. 

Sicily. 

Palmo,. 9.5 inches. 

Canna,.76. inches. 

Common salma,. 7.85 bushels. 

Gross salma,.... 9.77 bushels. 

Barile, { of a liquid salma,. 2.8825 gallons. 

Gross rottolo, y^ of a gross cantaro, ... 1.925 pound. 

Neat rottolo, T £y of a neat cantaro,. 1.75 pound. 

Pound,.7 pound. 

Malta. 

Palmo,.10.2375 inches. 

Canna,.81.9 inches. 

Salma of grain,. 8.221 bushels. 

Rottolo, of a cantaro,. 1.75 pound. 

26 

































302 


WEIGHTS AND MEASURES. 


Venice. 

Braccio for silks,..24.8 inches. 

Braccio for woollens,.. .26.61 inches. 

Staio, £ of a moggio,.2.27 bushels 

Bigoncia, £ of an anfora,.34.24 gallons. 

Gross pound,..... 1.0518 pound. 

Neat pound,.644 pound. 

Trieste. 

Braccio for silks,.25.2 inches. 

Braccio for woollens, .26.6 inches. 

Staio,. 2.344 bushels. 

Orna, or eimer,.15. gallons. 

Commercial pound,. 1.236 pound. 

Smyrna. 

Pic,..’... .27. inches. 

Killow,. 1.456 bushel. 

Oke,. 2.833 pounds. 

EAST INDIES. 

Mauritius, or the Isle of France. 

The weights and measures are those of England, and 
the old ones of France. 

Calcutta. 

Haut, or cubit, £ of a guz or yard,.18. inches. 

Coss, or mile,. 1.136 mile. 

Factory maund, .74§ pounds. 

Bazar maund,.82 T ^- pounds. 

Bombay. 

'Haut, or cubit,.18 inches. 

Candy,.25 bushels. 

Seer, T \j- of a maund, of a candy,.... 11^ ounce. 

Bag of rice, 6 maunds,.168 pounds. 

Madras. 

Haut, or cubit,.18. inches. 

Parah, ^ of a garce,. 1.75 bushel. 

Maund, ^ of a candy,.2.25. pounds. 


























WEIGHTS AND MEASURES. 


303 


Bencoolen, in Sumatra. 


Bamboo, ^ of a coyang,. 1 gallon. 

Bahar,.560 pounds. 

Batavia, in Java. 

Foot,...12.36 inches. 

Ell,.27. inches. 

Kann,..... 1.58 quart. 

Pecul,.135.625 pounds. 


The other weights and measures are like those of Hol¬ 
land. 

Canton. 


Covid, or cobre,. 

Catty,. 

Pecul,.. 

Japan. 

Inc, or tattamy,... 

Catty,. 

Pecul,. 


.14.625 inches. 

. 1£ pound. 
133£ pounds. 


6.25 feet. 

1.3 pound. 
130. pounds. 


WEST INDIES. 


British Islands. 

Weights and measures of Great Britain. 

French Islands. 

The old and new systems of French weights and meas¬ 
ures are used, and also the English wine gallon. 

St. Domingo, or Hayti. 

Old French weights and measures, also the English 
wine gallon. 

Danish Islands. St. John, St. Thomas, and Santa Cruz. 

Danish weights and measures, also the English foot and 
yard. 

Dutch Settlements. St. Eustatia, St. Martin, Curazoa, 
Surinam, Berbice, Demarara, and Essequibo. 
Weights and measures of Holland. In Curazoa, how- 














304 


WEIGHTS AND MEASURES. 


ever, they also employ the Spanish vara of 33.375 inches, 
and a pound weighing 1.17 pound. 

Spanish Islands. See Spain. 

Lesson 225. 

1. If 7,000 pounds of sugar, old French weight, import¬ 
ed from Guadaloupe, cost 6 cents a pound, delivered in 
New York, at what price must it be sold by the American 
pound to gain $184.80 on the whole amount ? 

Ans. at 8 cts. 

2. How many acres are there in a French myriare ? 

Ans. 247 A. 18 sq. rods 1934 sq. ft. 

3. A merchant sold 650 bushels of corn in Cumana, 

Colombia, at 10 reals a fanega ; what price in Federal 
money did he get a bushel ? Ans. $.78174, nearly 

4. If you purchase 500 almuides of wine in Lisbon, for 
1,311 Spanish dollars, how much a gallon does it cost you ? 

Ans. $.60. 

5. How many yards are there in 575 Hamburg ells ? 

Ans. 360.62, about. 

6. If 105 barili of wine are imported into St. John, New 

Brunswick, from Leghorn, and sold at 4s. 6d. Canada cur¬ 
rency, an imperial gallon, how much does the whole bring 
in dollars and cents ? Ans. $944.75. 

7. How many gallons are there in 25 Venetian anfora of 

wine ? Ans. 3,424, about. 

8. If you buy 9,000 pounds of hides in Port au Prince, 

St. Domingo, what number of American pounds will you 
obtain ? Ans. 9,720. 

9. A merchant imported 5,000 sheffels of wheat from 

Prussia, in 1837 ; how many bushels were there in that 
quantity ? Ans. 7,797. 

10. If I buy 100 bags of rice in Bombay, for 840 rupees, 
what price a pound do I give in Federal money ? 

Ans. .0225. 


BOOK-KEEPING. 


305 


BOOK-KEEPING. 


FARMERS’ AND MECHANICS’ METHOD. 
Lesson 226. 

The Farmers’ and Mechanics’ method of Book-Keeping 
should be employed by farmers and mechanics, and by all 
persons in their household expenses, or any small business 
in which they may be engaged. 

Accounts are kept by this method in a single book ; on 
page 1 are written the owner’s name, his place of resi¬ 
dence, the date, and a description of the book. If the 
book is to contain his general accounts, he writes Account 
Book for a description ; if the book is to contain accounts 
concerning his household expenses only, he writes House¬ 
hold Expenses ; if the book is to contain accounts con¬ 
cerning his farm in Medford only, he writes Accounts con¬ 
cerning Farm in Medford, &c. 

On pages 2 and 3, an account is opened with a certain 
person ; the things for which he is debtor are placed on 
the left hand page, and the things for which he is creditor 
are placed on the right hand page. On pages 4 and 5 an 
account is opened with another person in the same man¬ 
ner, and so on, always taking two pages for each person’s 
account. But if you are likely to have extensive dealings 
with any person, more than two pages should be left for 
his account. 

Example of Book-keeping by the Farmers’ and Mechan¬ 
ics’ method. 


[Page 1 .] 

JOSEPH SMITH. 

Cambridge, April 1, 1834. 

ACCOUNT BOOK. 

By whom should the Farmers’ and Mechanics’ method of Book- 
Keeping be employed ? 

In what are accounts kept by this method ? What is written on 
page 1 ? On pages 2 and 3 ? 4 and 5 ? 

How many pages do we always take for each person’s account? 
What if you are likely to have extensive dealings with any person ? 

26 * 





306 


BOOK-KEEPING. 


Page 2 . 

Dr. Alexander Dolland. 


1834. 


8 

c. 

April 5. 

To one day’s Work of myself and oxen, 

2 

25 

“ 19. 

“ 9 cwt. of Hay, at 95 cts. cwt., .... 

8 

55 

July 21. 

“ 6 bu. of Potatoes, at 45 cts. a bu., . 

2 

70 

Nov. 19. 

(t 3 cords of Wood, at $5.50 a cord, . 

16 

50 

1835. 




Jan. 6. 

“ 4 bu. of Corn, at $1.12£ a bu.,.... 

4 

50 


Page 4. 

Dr. William Clark, of Watertown, Innkeeper. 


1834. 


$ 

c. 

April 12. 

To 16 bu. of Potatoes, at 40 cts. a bu., 

6 

40 

July 3. 

“ 6 lbs. of Butter, at 22 cts. a lb., ... 

1 

32 

<( <( 

“ 9 doz. of Eggs, at 16§ cts. a doz., . 

1 

50 

(C It 

“ 8 lbs. of Cheese, at 10 cts. a lb., .. 


80 

Aug. 1. 

“ Horse to Boston,. 


40 



10 

42 




iwm 


Page 6. 

Dr. Jeremiah Dustwich. 


$ 


1834. 

May 2. 
June 14. 


To labor of my oxen half a day,. 
“ 7 lbs. of Lard, at 16 cts. a lb. 


• • 


1 


c. 

60 

12 

















BOOK-KEEPING. 


307 


Page 3. 

Alexander Dolland. 


Cr. 


1834. 

April 19. 

By 18 lbs. of Flour, at 4 cts. a lb.,.... 
“ 1 Plough,. 

$ 

c. 

72 

May 7. 

7 

50 

< c <( 

“ 2 Shovels, at $1.25 apiece,. 

2 

50 

Aug. 11. 

“ 1 bl. Flour,. 

8 

00 

Sept. 20. 

“ 7 lbs. of Black Tea, at 35 cts. a lb., 

2 

45 

( C (f 

“ Cash,..... 

4 

00 


Page 5 . 

William Clark, of Watertown, Innkeeper. Cr. 


1834. 

April 19. 
July 30. 

By Cash... 

$ 

5 

c. 

00 

“ 10 lbs. of Sugar, at 9 cts. a lb., ob¬ 
tained by an order from Wm. Clark 
on Joel Davis,. 


90 

Sept. 1. 

11 Cash to balance,.. 

4 

52 





$10 

42 


Page 7. 

Jeremiah Dustwich. 


Cr. 


1834. 

April 18. 


By shoeing my horse, 

























308 


BOOK-KEEPING. 


Great care should be taken to keep all the accounts in 
a uniform manner ; to write a plain and exact descrip¬ 
tion of every transaction on the first opportunity, before 
any part of it is forgotten. By taking this course, many 
losses, misunderstandings, disputes, and lawsuits may be 
avoided. 

When an account is settled, the amounts of the Dr. and 
Cr. columns are added up, and two heavy black lines 
drawn beneath. 

It is always well to state a person’s residence, if he lives 
out of town, and his occupation, if any question can be 
raised as to his identity. 

In the latter part of the book there should be an index. 

The following is an index to the preceding book. 


INDEX. 

Page 


A. 

B. 

C. Clark, William,.4 

D. Dolland, Alexander,.2 

Dustwich, Jeremiah,.6 

E. 


&c. 


What should we take great care to do ? What shall we avoid by this 
course ? 

What is done when an account is settled ? 

By the preceding account book. whose account appear to be settled ? 
Whose accounts appear unsettled ? 

What if a person lives out of town, or if a question can be raised as 
to his identity ? 

What should there be in the latter part of the book ? 







BOOK-KEEPING. 


309 


Lessons 227 and 228. 

How should Jonathan Belcher of Hartford prepare hi? 
book and keep his accounts, beginning January 1, 1830, 
if he buys of George Hodges, Jan. 6, 4 lbs. of tea at 33 
cts. a lb., 6 lbs. of sugar at 10 cts. a lb., and an iron ket¬ 
tle for $2 ; sells Abraham Adams, Jan. 16, 2 tons of hay 
at $17 a ton ; sells David Bartlett, Feb. 3, 11 bushels of 
corn at $1.10 a bushel, and 14 lbs. of butter at 16§ cts. 
a lb. ; buys a cart of John Farrell, Feb. 27, for $47, a 
plough for $10, and a horse for $67 ; pays John Farrell, 
March 9, $17 in cash, sells him two cows, one for $19 
and the other for $20 ; sells David Bartlett, March 9, 25 
bushels of potatoes at 30 cts. a bushel, and a quarter of 
veal, weighing 11 lbs., at 8 cts. a lb. ; delivers to John 
Farrell, March 12, an order on David Bartlett for $20, 
and charges Farrell with the order, giving Bartlett credit 
for it ; receives of Abraham Adams, March 15, $25 in 
cash, and sells him the same day, 16 bushels of oats at 
60 cts. a bushel, 5 bushels of corn at $1 a bushel, and 45 
lbs. of butter at 20 cts. a lb. ; buys of George Hodges, 
April 2, 1 bl. of flour for $7.50, 14 lbs. of rice at 5 cts. a 
lb., 3 lbs. of tea at 30 cts. a lb., and 2 gals, of molasses 
at 35 cts. a gal. ; works, together with his oxen, 4 days, 
April 2, 3, 5, and 6, for Lemuel Snow, at $2.50 a day ; 
receives, April 9, of Lemuel Snow, $5 in cash, and a 
barrel of cider worth $2 ; employs Pliny Shaw, April 10, 
to shoe his horse for $1.33^, and to perform other black¬ 
smith’s work to the amount of 97 cts. ; works 2 days, 
April 12 and 13, together with his oxen, in moving Robert 
Sampson, of East Hartford, at $2.50 a day ; pays George 
Hodges, April 14, $10 in cash ; lets Lemuel Snow have 
his oxen 3 days, April 15, 16, and 17, at $1.50 a day ; 
buys of George Hodges, April 27, 7 lbs. of coffee at 16 
cts. a lb., 12 lbs. of salt fish at 4 cts. a lb., and 2 doz. of 
crackers at 10 cts. a doz. ; receives of Robert Sampson, 
April 30, $5 in cash, and sells him the same day 45 lbs. 
of cheese at 10 cts. a lb., and 8 doz. of eggs at 14 cts. a 
doz.; May 1, employs Pliny Shaw to repair his plough for 
$1, and to new steel his axe for 75 cts.; hires Davis 
Weed, May 6, 7, and 8, to paint his house, at $1.75 cts. a 
day, and buys the paint of him for $18 ; hires John Smith, 


310 


BOOK-KEEPING. 


carpenter, May 7 and 8, to work on his house, at $1.62£ a 
day ; pays David Weed, May 12, $5 in cash, and John 
Smith, carpenter, $3.35 ; sells Abraham Adams, May 20, 
4 pigs at $2 apiece, and receives of him that amount in 
cash ; pays John Farrell, May 25, $10 in cash ; hires 
John Smith, carpenter, to work on his shed 5 days, May 
27, 28, 29, 31, and June 1, at $1.50 a day ; sells Lemuel 
Snow, June 1, a calf for $5.75, and receives of Robert 
Sampson $4.75 in cash ; settles with John Farrell, June 
10, paying him his due in cash ; buys of George Hodges, 
June 15, 11 lbs. of sugar at 10 cts. a pound., 5 gals, of 
molasses at 33^ cts. a gal., and 12 yds. of cotton cloth at 
22 cts. a yd. ; sells John Farrell, June 16, 7 bushels of 
corn at $1 arbushel, and 5 bushels of rye at $1.30 a bushel. 

Note. To record the preceding transactions, double 2 £ sheets of let¬ 
ter paper, thereby making a short but wide book of 20 pages. 


BOOK-KEEPING BY SINGLE ENTRY. 

Lesson 229. 

Retail merchants and traders with a moderate business, 
keep their accounts in a way somewhat like that which we 
have described. The original charges, however, are made 
in what is called a day book , where they are written one 
after another, in the order in which the transactions occur. 
During the hours of leisure, these charges are copied into 
another book, arranged just like that in lesson 226, the ac¬ 
count of each man being placed under his name. This 
book is called the leger. The act of copying from the day 
book into the leger is called posting. Merchants use the 
day book, because it is inconvenient, during the hours of 
business, to turn over the leger and search out each cus¬ 
tomer’s name in order to record the trade they make with 
him. 


How do retail merchants and traders with a moderate business keep 
their accounts ? In what are the original charges made ? How are 
they written? Into what are these charges copied, and how is the ac¬ 
count of each man placed ? What is this book called ? What is the act 
of copying the day book into the leger called ? Why do merchants use 
the day book ? 




BOOK-KEEPING. 


311 


In case of a dispute or suit, the day book, or the book in 
which the original charges are made, is the only one to be 
examined or produced. The charges in the leger ought to 
be exact copies of those in the day book, unless the busi¬ 
ness be extensive, when it is customary to abridge them in 
posting, to save labor and room. 

As soon as an account is posted, we mark it, by writing in 
the first column of the day book, opposite the account, the 
page of the leger to which the account has been transferred; 
and in the first column of the leger, opposite the account 
the page of the day book where the account is to be found. 

When the day book is filled, we can commence another 
and continue on, but when the leger is filled, we must add 
up the Dr. and Cr. columns of all the unsettled accounts, 
and transfer the amounts to the top of the Dr. and Cr. 
columns of the same persons in the new ledger. The first 
day book is called day book A, the second, day book B, 
and so on ; the first leger is called leger A, and the second, 
leger B, and so on. 

This method of keeping accounts with a day book and 
leger, is called Book-Keeping by single entry. 

Example of Book-Keeping by Single Entry. 

DAY BOOK. 


Page 1. 

JOHN REED, 
Portsmouth, June 3, 1830. 
DAY BOOK. 


What book is the only one to be examined or produced in case of a 
dispute or suit ? Of what ought the charges in the leger to be exact 
copies? What, however, is customary when the business is extensive ? 

What is done as soon as an account is posted ? 

What is done when the day book is filled ? When the leger is filled ? 
What are the first, second, &c., day books called ? Legers ? 

What is this method of keeping books with a day book and leger 
called ? 




312 


BOOK-KEEPING. 


Page 2 . 

Portsmouth, June , 1830. 


Reuben Farnham,. Dr. 

To 1 bl. Pork,.$25.00 

“ 6 gals. Molasses, at 35 cts. a gal., 2.10 

“ 6 yds. Calico, at 28 cts. a yd., .... 1.68 


David Brown, Machinist,. Dr. 

To 1 piece Sheeting, 27f yds., at 18 cts. a yd., 
-June 5.—- 


Oliver Dana, ofKittery,. Cr. 

By 12 lbs. Butter, at 16§ cts. a lb.,. ... $2.00 
“ 4 doz. Eggs, at 12^- cts. a doz.,... 50 


Dr. 


To 14 lbs. Flour, at 64 cts. a lb. 


28 

5 


78 

00 

50 

91 


Page 3 . 

Portsmouth, June 7, 1830. 


Thomas Avery, . Dr. 

To 56 lbs. Loaf Sugar, at 18 cts. a lb.,.. 


William Dyer,. 

By 2 hhds. Molasses, at $28 a hhd., ... 
; 5 chests Black Tea, at $16 a chest, 


10 To Cash, 


-June 8.- 


Samuel Everett,. 

12 To 16 lbs. Nails, at 9 cts. a lb., .. 

13 By 5 bu. Rye, at $1.25 a bu., ... 
“ 4 c °rd of Wood, at $5 a cord, 


Page 4 . 

Portsmouth, June 9, 1830. 


Cr. 

$56.00 

80.00 

Dr. 


Dr. 

Cr. * 
$6.25 
2.50 


10 

136 

45 


00 

00 


1 44 


8 75 


David Brown, Machinist,. 

... Cr. 

$ 

c. 

By Labor, mending safe,.. 


“ 1 Lock,.. 




“ Cash, .. 






„ 5 

00 

Account settled. 

















































BOOK-KEEPING. 


313 


-June 10.- 


Thomas Avery, . Dr. 

To 20 quintals Cod-fish, at $3.80 a quin¬ 
tal, .$76.00 

“ 3 pieces Brown Sheeting, 81 yds., 

at 14 cts. a yd.,. 11.34 


Cr. 


By Cash, .. 

Mr. Avery agrees to pay the balance of his 
account in 90 days, in cash, 

---June 11.- 


Reuben Farnham,. Cr. 

By Cash,. 


15 


34 

00 


00 


Page 5 . 

Portsmouth, June 12, 1830. 


Reuben Farnham,. 

2 To 16 lbs. Cheese, at 10 cts. a lb.,.... $1.60 
8 lbs. Black Tea, at 35 cts. a lb., .. 2.80 


Oliver Dana, of Kittery,. Dr.' 

To 5 yds. Broadcloth, at $4.50 a yd.,.. $22.50 
“ 1 Hat,. 4.75 


-June 14.- 


Oliver Dana, of Kittery,. Cr. 

By 15 bu. Corn, at 95 cts. a bu.,. 

-June 15.--- 


Reuben Farnham, . Dr. 

To 1 tub Butter, 42 lbs., at 20 cts. a lb. ...... 

Butter and tub weigh 46 lbs., tub is 
estimated to weigh 4 lbs. 

Mr. Farnham agrees to pay what is due on 
his account in 10 days, in Corn at $1 a bu. 


27 

14 

8 


40 

25 

25 

40 


Lesson 230. 

EXAMPLE OF A LEGER. 


Page 1. 

JOHN REED, 
Portsmouth, June 3, 1830. 
LEGER. 


27 






































S14 


BOOK-KEEPING. 


Page 2. 

Dr . Reuben Farnham. 



1830. 


$ 

c. 

2 

June 4. 

To 1 bl. Pork,.$25.00 





“ 6 gals. Molasses, at 35 cts. a gal., .... 2.10 





“ 6 yds. Calico, at 28 cts. a yd.,.1.08 





— 

28 

78 

5 

“ 12 . 

M 16 lbs. Cheese, at 10 cts. a lb.,.1.60 





“ 8 lbs. Black Tea, at 35 cts. a lb.,.2.80 





— 

4 

40 

5 

“ 15. 

“ 1 tub Butter, 42 lbs. at 20 cts. a lb.,. 

8 

40 



Page 4. 



Dr. 

David Brown, Machinist. 




1830. 


$ 

c. 

2 

June 4. 

To 1 piece Sheeting, 27| yds., at 18 cts. a yd., . 

^5 

00 




{gauss 

“ 

■ 


Page 6 . 



Dr. 

Oliver Dana, of Kittery. 




1830. 


$ 

C. 

2 

June 5. 

To 14 lbs. Flour, at 6 <| cts. a lb.,. 


91 

5 

“ 12 . 

“ 5 yds. Broadcloth, $4.50 a yd.$22.50 





“ 1 Hat,.4.75 





• 

27 

25 



Page 8 . 



Dr. 

Thomas Avery. 





1830. 


$ 

c. 

3 

June 7. 

To 56 lbs. Loaf Sugar, at 18 cts. a lb .. 

10 

08 

4 

“ 10. 

“ 20 quintals Cod-fish, at $3.80 a quintal, $76.00 





“ 3 pieces Brown Sheeting, 81 yds., at 





14 cts. a yd.,.11.34 




, 


87 

34 



Page 10. 


' 

Dr. 

William Dyer. 




1830. 


$ 

c 

3 

June 7. 

To Cash,. 

45 

00 


1 

1 

Page 12. 


Dr. 

Samuel Everett. 




1830. 


$ 

, c - 

3 

June 8. 

To 16 lbs. Nails, at 9 cts. a lb.,. 

1 

44 












































BOOK-KEEPING. 


315 


Page 3. 

Reuben Farnham. 


Cr. 


1830. 


$ 

June 11. 

By Cash,. 

15 




Page 5. 

David Brown, Machinist. 


Cr. 



1830. 



$ 

c. 

4 

June 9. 

By Labor, mending safe,. 





“ 1 Lock,. 






“ Cash,. 







— 

5 

00 






— 



Page 7. 






Oliver Dana, of Kittery. 


Cr. 


1830. 1 


1 $ 

c. 

2 

June 5. 

By 12 lbs. Butter, at 16§ cts. a lb., . . . . 
u 4 doz. Eggs, at 12£ cts a doz., . . . . 

. $2.00 





. . 50 

2 

50 

5 

14. 

“ 15 bu. of Corn, at 95 cts. a bu., . . . . 


14 

25 


Page 9. 

Thomas Avery. Cr. 



1830. 


$ 

! c. 

4 

June 10. 

By Cash,. 

50 

00 



Page 11. 





William Dyer. 

Cr. 


1830. 


$ 

c. 

3 

June 7. 

By 2 hhds. Molasses, at $28 a hhd.,.... $56.00 





“ 5 chests Black Tea, at $16 a chest, . . . 80.00 





— 

136 

00 



Page 13. 





Samuel Everett. 

Cr. 


1830. 
June 8. 

By 5 bu Rye, at $1.25 a bu., .... 


$ 

c. 


“ £ cord Wood, at $5 a cord., . . . 


8 

75 











































316 BOOK-KEEPING. 

There should be an index to the leger precisely like one 
to an account book. 

The following is an index to the preceding Leger. 

Tage. 


A. Avery, Thomas,. B 

B. Brown, David,. 4 

C. 

D. Dana, Oliver, . 6 

Dyer, William,. 10 

E. Everett, Samuel, . 12 

F. Farnham, Reuben, . 2 

G 

&c. 


Those who deal much in money, have a small book in 
which the cash received and paid, is recorded by itself. 
This book is called a cash book. 

Example of a Cash Book formed from the preceding day 
book. 


CASH BOOK. 


Cash. 


Received. 

1830 $ 

June 9, Of David Brown, 3.00 
“ 10, “ Thomas Avery, 50.00 
“ 11, “ Reuben Farnham, 15.00 


Paid. 

1830 

June 7. To Wm. Dyer, 


$ 

45.00 


Note. We have stated that many persons abridge the day book in 
posting. Several things bought or sold the same day, they call sundries 
in the leger; when only one article is bought, or sold the same day, it 
is named, and cash is always named ; thus David Brown’s account, in 
the preceding leger, would stand as follows; 


Dr. David Brown. || David Brown. Or. 


18:50 


$|c. 1 


1830 


$ 

June 4. 

To Sheeting, 

jjljB 

4 

June 9. 

By Sundries, $2.00 
“ Cash, 3.00 

5 


This method of abridging saves some labor in posting ; besides much 
more can be written in the leger than by the other course. It should 
be adopted when the business is extensive. 


Whose account in the preceding leger is marked as being settled ? 
What is said of an index ? 

What do those who deal much in money have ? What is,it called ? 
How do some persons abridge the day book in posting? 

What does this method of abridging save ? When should it be 
adopted ? 





















BOOK-KEEPING. 


317 


The advantages of copying every charge in full into the leger, are as 
follows; any man’s account can be drawn off with ease, without turn¬ 
ing over the day book, and copying out the items of the account from 
perhaps ten or twenty different places; a new clerk, or a person but 
little skilled in Book-Keeping, can understand a leger so kept at a 
glance; a man can be shown the state of his account, and be satisfied 
with regard to each item of it without opening the day book, and expos¬ 
ing other people’s accounts on the same pages as his own, showing him, 
perhaps, that some person has obtained an article for a smaller price 
than himself; also, if the day book is carried out to be used in a settle¬ 
ment, or a suit, the accounts can be kept in the leger during the inter¬ 
val, and afterwards, be copied into the day book. 

Lessons 231 j 232, and 233. 

Prepare a day book, leger and cash book, for Moses 
Harris of Lowell, and record in them the following trans¬ 
actions. 

January 2, 1833, Mr. Harris bought of Cyrus Neal of 
Boston, 2 pieces of broadcloth, containing 56 yds., at 
$3.50 a yd., 6 pieces of sheeting, containing 159 yds. at 15 
cts. a yd., 3 pieces of shirting containing 82 yds., at 14cts. 
a yd., and 50 yds. of calico, at 20 cts. a yard. ; bought of 
David Pierce of Boston, 27 hats, at $3.15 a piece, and 16 
pr. of boots at $3 a pr. ; bought of Henry Gray of Boston, 
3 hhds. of Molasses at $26 a hhd., 8 bis. of flour at $7.50 a 
bl., and 4 chests of green tea at $24 a chest, and paid 
Gray $100 in cash ; Jan. 3, sold Tobias Cary of Tewks¬ 
bury, 18 lbs. of feathers at 52 cts. a lb., and 26# yds. of 
shirting at 18 cts. a yd., and received of him 15 bushels of 
corn at 98 cts. a bushel ; bought of Joshua Higgins 250 
lbs. of cheese at 8 cts. a lb., paid Higgins $10 in cash, and 
agreed that he should take the rest of his pay in goods ; 
Jan. 5, sold Henry Gray of Boston, 800 lbs. of butter, at 
18 cts. a lb., 100 lbs. of cheese at 10 cts. a lb., and 20 doz. 
of eggs, at 20 cts. a doz. ; bought of Silas Strong, 4 cords 
of oak wood at $5 a cord, and sold him 5 yards of broad¬ 
cloth at $4.50 a yd. ; Jan. 7, sold Joshua Higgins 1 hat 
for $4 50, 4 gals, of molasses, at 37£ cts. a gallon, and 2 
lbs. of coffee at 18 cts. a lb., and bought of him, 16 doz. of 
eggs at 14 cts. a doz. ; paid Cyrus Neal of Boston, $80 in 
cash ; bought of Zebediah Burney of Boston, 400 lbs. of 
iron at 7 cts. a lb., 2 casks of nails, containing 225 lbs. a 


What are the advantages of copying each charge in full, into the 
leger ? 


27* 



318 


BOOK-KEEPING. 


piece, at 7£ cts. a lb., 25 brooms, at 14 cts. a piece, and 
12 axes, at $1.25 a piece ; bought of Henry Gray of Bos¬ 
ton, 60 bushels of salt at 40 cents a’bushel, 1 hhd. of 
brown sugar, weighing 7 cwt. 3 qrs. 13 lbs., at 7 cts. a lb., 
6 bis. of flour at $7 a bl., and 45 lbs. of raisins at 10 cts. 
a lb. ; Jan. 8, sold John Williams, carpenter, 42 lbs. of 
nails at 9 cts. a lb., 1 broad-axe for $2, and 16 lbs. of 
cheese at 12 cts. a lb. ; sold Thomas Gurney 1 piece of 
shirting containing 27£ yds., at 19 cts. a yd., and a pair of 
boots lor $4, tind received of him, $5 in cash ; Jan. 9, 
sent Henry Gray of Boston, $75 by Edward Stickney ; 
sold Tobias Cary of Tewksbury, 8 lbs. of brown sugar at 
10 cts. alb., 4 lbs. of green tea at 38 cts. a lb., 2 gals, of 
molasses at 36 cts. a gal., and 6 lbs. of raisins at 16 cts. a 
lb. ; sold Thomas Gurney 1 iron bar, weighing 15 lbs., at 
10 cts. a lb., 1 stone hammer, weighing 12 lbs., at 16§ cts. 
a lb. ; bought of Jonas Howe of Dracut, 8 bis. of apples 
at $2.50 a bl., 25 bushels of rye, at $1 a bushel, and 14 
bushels of corn at 95 cts. a bushel ; Jan. 10, received of 
Tobias Cary of Tewksbury, $2 in cash ; bought of Jonas 
Howe of Drucut, 11 bushels of beans at $1.50 a bushel, 
and sold him 1 axe for $1.75, 2 pr. of woollen mittens, at 
37£ cts. a pr. 6 pr. of woollen stockings at 41§ cts. a pr., 
and 1 pr. of pantaloons for $4.75 ;^Jan. 11, paid David 
Pierce of Boston, $100 in cash ; bought of Cyrus Neal of 
Boston, 1 bale of sheeting, containing 400 yds., at 14 cts. 
a yd., and 2 pieces of blue cassimere, containing 21 yds. a 
piece, at $1.16§ a yd. ; Jan. 12, bought of Joshua Hig¬ 
gins 6 hogs, the total weight of which was 1,584 lbs., at 9 
cts. a lb., and sold him 3 bis. of flour at $8 a bl., 112 lbs. 
of rice at 5 cts. a lb., 1 bl. of molasses, containing 30 
gals., at 33£ cts. a gal., and 1 quintal of cod fish for $4 ; 
Jan. 14, sold Henry Gray of Boston, 9 bis. of pork at $22 
a bl. ; bought of Tobias Cary of Tewksbury, 4 of a pig, 
weight 118 lbs. at 10 cts. a lb., and 16 bushels of potatoes 
at 38 cts. a bushel ; bought of Silas Strohg, 25 lbs. of but¬ 
ter at 16 cts. a lb., and sold him 18 lbs. of salt fish at 4 
cts. a lb. ; Jan. 15, bought of Silas Strong a horse for $75, 
and paid the cash for him ; sold Thomas Gurney 75 lbs. 
of cast steel at 12£ cts. a lb., and 100 lbs. of iron at 5 cts. 
a lb. ; Jan. 16, sold Jonas Howe of Dracut, 8£ yds. of 
cassimere at $1.25 a yd., and 2 yds. of shirting at 18 cts. a 
yd. ; sold Tobias Cary of Tewksbury, 2 bis. of flour at 


BOOK-KEEPING. 


319 


88.50 a bl., and settled ; Jan. 17, sold John Williams, car¬ 
penter, a firkin of butter at 23 cts. a lb., the butter and 
tub weigh 58 lbs., and the tub is estimated to weigh 5 lbs., 
the tub is to be returned and the weight rectified ; Jan. 18, 
bought of Cyrus Neal of Boston, 1 piece of broadcloth, 
containing 29 yds., at 84 a yd., and 2 bales of shirting, con¬ 
taining 400 yds., at 14 cts. a yd. 

JVote. To record the preceding transactions, double 1 sheet of letter 
paper, thereby making a short but wide day book -of 8 pages. Also 
double 3 sheets of letter paper, thereby making a short but wide leger 
of 24 pages. In the leger on the last page but two, write the index, 
and on the last page but one, write the cash book. The charges should 
also be posted into another leger of the same size, by the abridged 
method. 


OBSERVATIONS. 

If a merchant takes an exact inventory or account of his 
unsold goods and of all the rest of his property, including 
the debts owed to him, and deducts from this sum the 
amount of the debts he owes, he will obtain the present 
value of his property. This, compared with the value of 
his property at a certain time before, shows what he has 
gained or lost since that time. 

Book-Keeping by single entry is unfit for merchants 
engaged in extensive business ; they usually keep their 
accounts by a system called double entry. A description 
of this system requires a treatise ; a short account of it, 
like those appended to some arithmetics, is quite useless 
and insignificant. All persons should understand Book- 
Keeping by the methods which we have explained. Book- 
Keeping by double entry is necessary for a few only. 


How can a merchant find the present value of his property, and his 
gain or loss since a certain time ? 

What is Book-Keeping by single entry unfit for? How do they usual¬ 
ly keep their accounts ? What is said of Book-Keeping by double entry ? 
What is said of the necessity of understanding Booa-Keeping by the 
different methods ? 



BUSINESS FORMS. 


BUSINESS FORMS. 


Lesson 234. 

BILLS. 

No. 1. 

Form of a Common Bill. 

Mr. Joseph Snow 

to James Thompson,... .Dr, 
Feb. 7, 1837. To 11 bu. of Potatoes, at 38 cts. a bu., $4.18 
“ 15, “ “ 7 lbs. of Butter, at 20 cts. alb., ...1.40 

Mar. 4, “ “ 2 days’ Labor, at $1.25 a day,... .2.50 

$8.08 

April 3, 1837. Received payment. 

James Thompson. 

No. 2. 

Form of a Bill, taken from a Merchant’s Books. 

See Reuben Farnham’s Account in the preceding leger. 
Mr. Reuben Farnham 

to John Reed,.Dr. 

1830. 

June 4. To 1 bl. Pork,.$25.00 

“ 6 gals Molasses, at 35 cts. a gal., ..2.10 
“ 6 yds. Calico, at 28 cts. a yd.,.... 1.68 


« << 
it it 


“ 12 , 

<< {< 


15, 


16 lbs. Cheese, at 10 cts. a lb.,.... 1.60 
8 lbs. Black Tea, at 35 cts. a lb., .2.80 


1 tub Butter, 42 lbs., at 20 cts. a lb., 

Cr. 


“11, By Cash,. 

“ 21, ** Cash to balance, 


.26 

John Reed. 


78 

40 

40 

58 

00 

58 


Portsmouth, June 21, 1830. 











BUSINESS FORMS. 


321 

Observations. If Paul Dow, agent for James Thomp¬ 
son, receives the pay of bill No. 1, he acknowledges the 
receipt of the money thus ; 

Received payment, 

For James Thompson, 
Paul Dow. 


If Samuel Short, clerk or agent of John Reed, settles 
bill No. 2, he signs it thus ; 


John Reed, 
By Samuel Short. 


Note. A bill should not be signed until payment is received, or & 
settlement effected. 

1. If John Carter works 3 days, January 3, 5, and 6, 
1835, for Ephraim Wheeler, at $1.16§ a day ; sells Mr. 
Wheeler 18 cwt. of hay, Feb. 2, at 95 cts. a cwt. ; sells 
him a pig for $11, Feb. 17, and mends a plough for him, 
March 3, for which he expects 75 cts., how should he 
make out his bill, and how should his son, Walter Carter, 
acknowledge the receipt of the money, March 7, 1835 ? 

2. How should Thomas Cole, clerk of John Reed, make 
out Oliver Dana’s bill from the preceding leger, and settle 
it, June 23, 1830 ? 

3. How should John Reed make out the bill of William 
Dyer, from the preceding leger, before settlement, June 
25, 1830 ? How after settlement ? 

4. David Wilkins, on examining his books, finds Simp¬ 
son &, Clark charged with 11 bushels of beans, at $1.75 
a bushel, delivered August 15, 1831 ; 7 cwt. of hay, at $1 
a cwt., delivered Sept. 1; 48 bushels of potatoes, at 30 cts. 
a bushel, delivered Oct. 29 ; 16 bushels of corn, at $1.12£ 
a bushel, delivered Jan. 2, 1832 ; and he finds them 
charged $5 for breaking his cart, Jan. 3, 1832 ; how 
should he make out his bill, and acknowledge the pay¬ 
ment, Feb. 1, 1832 ? 

Explanation. The last item may be put in thus ; To 
breaking cart $5. 

5. How should Joseph Smith make out the bill of Jere¬ 
miah Dustwich, from the preceding account book, and 
acknowledge the receipt of what is due, July 5, 1834 ? 


What is said of signing a bill ? 



BUSINESS FORMS. 


Lesson 235. 

DUE BILLS. 

Trenton, December 4, 1826. 

Due John James, 5 dollars. 

Henry Sargent 

Note. Due Bills are only memorandums given for small sums. 

1. How should a due bill for $4.50, from Edward Jones 
of Auburn, to Calvin Stoop, be written, June 15, 1832 ? 

2. How should a due bill for $8.37, from William Brown 
of Portland, to Sylvester Freeman, be written, March 4, 
1836 ? 


NOTES. 

No. 1. 

Note on Demand. 

$50 Providence, January 3, 1838. 

For value received, I promise to pay John Davis, or 
order, fifty dollars on demand, with interest. 

Witness, Abram Horne. Benj. Homans. 


No. 2. 

Note Payable to Bearer. 

$100 Louisville, May 3, 1836. 

For value received, I promise to pay Simeon Fowle, or 
bearer, one hundred dollars on demand, with interest. 

Eben. Earle. 

No. 3. 

Note by Two Persons. 

$200 Portsmouth, July 15, 1838. 

For value received, we jointly and severally promise to 
pay Caleb Callahan, or order, two hundred dollars on 
demand, with interest. 


What are due bills ? 


Samuel Slater. 
Darius Bird. 




BUSINESS FORMS. 


323 


No. 4. 

«Note payable after a certain time. 

$127.33 Brooklyn, May 23, 1837. 

For value received, I promise to pay Cyrus Jackson, or 
order, one hundred and twenty-seven dollars and thirty- 
three cents, six months after date. 

Daniel Williams. 


No. 5. 

Note at Bank. 

$300 Boston, June 1, 1838. 

Ninety days from date, I promise to pay Timothy 
Stearns, or order, three hundred dollars, for value re¬ 
ceived. 

Hiram Hobart. 

Note. Timothy Stearns indorses the note, that is, writes his name 
on the back of it, thereby binding himself to pay it if Hobart is unable, 
and Hobart then proceeding to the bank with the note obtains the 
money. 

OBSERVATIONS ON NOTES. 

Unless the note has the words or order, or the words or 
bearer , it is not negotiable, that is, it cannot be sold, ex¬ 
changed, or traded with, because it is payable to only 
one person. 

If a note is payable to a certain person or order , say to 
John Davis or order, see No. 1, then John Davis can sell 
this note to whom he pleases, provided he indorses it, 
thereby binding himself to pay the note if the purchaser 
is unable to collect it of the signer, Benj. Homans. 

Several persons sometimes indorse a note, then each 
indorser is a security for the payment of it. 

If a note is payable to a certain person or bearer , say 
to Simeon Fowle or bearer, see No. 2, then Simeon Fowle 


By the preceding note at bank, how is the money obtained ? 

What if a note has not the words or order, or the words or bearer , 
on it ? 

If a note is payable to a certain person, or order, say to John Davis or 
order, on what terms can John Davis sell it? 

What if several persons indorse a note ? 

If a note is payable to a certain person, or bearer, say to Simeon 
Fowle, or bearer, to whom can Simeon Fowle sell it, and of whom can 
the purchaser collect it? 



3&4 


BUSINESS FORMS. 


can sell it to whom he pleases, and the purchaser can col¬ 
lect it only of Eben. Earle, the signer. 

A note payable at a certain time, is on demand on or after 
that time ; and if a note mention no time of payment, it 
is always on demand, whether the words on demand be ex¬ 
pressed or not. 

If no mention is made of interest, a note on demand 
draws interest as soon as payment is demanded ; and a 
note payable at a certain time, draws interest after that 
time. 

It is unnecessary to mention the rate, unless a rate lower 
than that fixed by law be agreed upon. 

If several persons sign a note promising jointly and sev¬ 
erally to pay a certain sum, like No. 3, this sum can be 
collected of either of the persons. 

If a note is payable within a certain time, in a particular 
article, say wheat, the holder is not obliged to receive the 
wheat after the expiration of the time, but can then de¬ 
mand money. 


Lesson 236. 

1. Write a note, dated Hartford, April 29, 1837, signed 
by Charles Lincoln, promising to pay Caleb Hunt, or 
order, $80 on demand, with interest. 

2. Write a note, dated Taunton, June 3, 1838, signed 
by Ezekiel Goodhue, and witnessed by Jonathan Tyler, 
promising to pay Ezra Dean, or bearer, $251.42, in one 
year from date. 

3. Write a note, dated Cincinnati, October 17, 1831, 
signed by John Sargent, E. Tucker, and Alfred Shaw, 
promising jointly and severally to pay Baird 8c Bowman, 
or order, $500 on demand, with interest. 

4. Write a note, dated New York, December 18, 1829, 
indorsed by Horace Smith and Warren Davis, and signed 
by James Green, promising to pay Horace Smith, or 


When is a note payable at a certain time, on demand? 

What if a note mention no time of payment ? 

If no mention is made of interest, when does a note on demand draw- 
interest, and when does a note payable at a certain time draw interest ? 
Is it necessary to mention the rate of interest ? 

What if several persons sign a note ? 

If a note is payable within a certain time, in a particular article, say 
wheat, when can the holder demand money ? 



BUSINESS FORMS. 


825 


order, at the Merchants’ Bank, $2,500 in sixty days from 
date. 

5. Write a note, dated Pittsburg, August 7, 1835, sign¬ 
ed by David Wilder, and witnessed by Alfred Davidson, 
promising to pay John Stone, or order, $1,000 eight 
months from date. 

6. Write a note, dated Bangor, February 15, 1831, 
signed by Abraham Nichols, promising to pay Benjamin 
Murdock, or bearer, $200 on demand, with interest. 


ORDERS. 


No. 1. 


Order for Money. 


Lowell, June 5, 1830. 

Mr. John Ryan, 

Please pay Mr. Phineas Stone, or order, twenty- 
five dollars on my account. 

Alexander Bernard. 


No. 2. 

Order for Goods. 

Hallowell, April 15, 1833 

Messrs. Swan, Porter Sc Co., 

Please pay Mr. Julian Long, or order, forty-five 
dollars in goods, and charge the same to me. 

George Crosby. 

Observations on Orders. When an order is made pay¬ 
able to a certain person or order, like the preceding, this 
person can sell it by indorsing it, thereby making himself 
security for the payment. An order made payable to a 
certain person or bearer , can be sold like an article of mer¬ 
chandise, without any indorsement, and the holder can 
collect it. Making an order is called drawing an order. 

7. How should Philip Barber of Natchez, draw an order 
on Isaac Shaw, June 29, 1835, directing him to pay John 
Black, or order, $35 on account of said Barber ? 


When an order is made payable to a certain person, or order, how 
can it be sold ? How can an order, made payable to a certain person, 
or bearer, be sold and collected ? What is making an order called ? 


28 




326 


BUSINESS FORMS. 


8. How should Goodwin & Clapp of New York, draw 
an order on Cyrus Holman, December 2, 1837, for $100, 
in favor of James Cook, or bearer. 

9. How should Hall & Rogers of Providence, draw an 
order on Samuel Perry, February 14, 1825, directing him 
to pay George Rice, or order, $180 in goods ? 


Lesson 237. 

RECEIPTS. 

No. 1. 

Receipt for Money on a Mote. 

If Samuel Reed owes John Blake of Boston, a note for 
$100, and pays him $15, it is customary to write a receipt 
on the back of his note, thus ; 

Boston, March 20, 1835. N Received fifteen dollars. 

If the note be not at hand, a receipt is made in this 
way ; 

Boston, March 20, 1835. 

Received of Samuel Reed fifteen dollars on his note for 
one hundred dollars, dated August 15, 1834. 

John Blake. 


No. 2. 

Reeeipt for Money on Account. 

Charleston, July 7, 1838. 

Received of Solomon Jenkins ninety-four dollars on 
account. Ezra Blair. 


No. 3. 

Receipt for Money obtained on an Order. 

The receipt is indorsed on the order thus ; 

Utica, January 15, 1835. 

Received of Mr. Silas Derby, twenty dollars, the amount 
of the within order. John Dory. 



BUSINESS FORMS. 


327 


No. 4. 

Receipt for Goods received on an Order. 

The receipt is indorsed on the order thus ; 

Albany, October 1, 1831. 

Received of Charles Quackenboss, eighty dollars in 
goods, the amount of the within order. 

Thomas Kelder. 

Note. When a part only of an order is paid, the sum can be indorsed 
on the order the same as the receipt on the back of a note. 

No. 5. 

General Receipts. 

Louisville, November 18, 1830. 

Received of Thomas Miller twenty-five dollars in full 
of all accounts. John Partridge. 

Louisville, November 18, 1830. 

Received of John Partridge twenty-five dollars in full 
of all accounts. Thomas Miller. 

Observations on Receipts. Receipts in full of all accounts , 
like the preceding, cut off accounts only, but if they are 
given in full of all demands , they cut off all accounts, and 
all demands of every kind. Such receipts are usually ex¬ 
changed between persons who have had business with 
each other when a final settlement is made. 


No. 6. 

Receipt for Money paid before it is due. 

Boston, May 6, 183Q(f 

Received of Joseph Hurd two hundred and fifty dollars, 
in full for the rent of my house from April 1, 1836, to April 
1, 1837. Jacob Parker. 

1. Write a receipt on the back of a note for $5, paid in 
Worcester, October 7, 1837. 

2. Write a second receipt on the back of the same note 
for $13, paid in Worcester, January 15, 1838. 

3. How should Joshua Day write a receipt for $81, paid 
him by Jonas Bridge in Troy, March 4, 1837, Day having 


What if a part only of an order is paid ? 

What is said of receipts in full of all accounts ? In full of all demands ? 
When are such receipts usually exchanged ? 



328 


BUSINESS FORMS. 


a note against Bridge for $300, dated July 6, 1836, the 
note not being at hand ? 

4. How should Henry Gay write a receipt for $75, paid 
him by Samuel Drake in Concord, July 16, 1838; Gay 
having a note against Drake for $150, which note was not 
at hand ? 

5. How should Edward Clough of Boston, write a re¬ 
ceipt for $50, paid him on account by David Clark, Au¬ 
gust 21, 1837, in Albany ? 

6. Ezekiel Snow of Newbury, has an order on John 
Hermann for $80 ; if he is paid August 15, 1831, how and 
where should he write a receipt ? 

7. Mark Doyle of Norwich has an order on Walter 
Lewis for $45 in goods ; if he is paid February 5, 1831, 
how and where should he write a receipt ? 

8. If Ezekiel Snow, in example 6, received only $30, 
August 15, 1831, how and where should he write a receipt ? 

9. How do George Perley and Stephen Guild of Har¬ 
risburg write receipts in full of all accounts to be ex¬ 
changed February 28, 1830 ? 

10. Write receipts for $10 in full of all demands, to be 
exchanged between Daniel Wood and James Stockton, in 
St. Louis, April 13, 1832. 

11. Write a receipt for $95, paid in Hudson, June 20, 
1832, by Joseph Doane to Frederick Swift, for the yearly 
rent of a farm ; the year ending June 1, 1834 ? 


Lesson 238 and 239. 

DRAFTS AND BILLS OF EXCHANGE. 

No. 1. 

Form of a Draft. 

$750^^- New York, August 11, 1837. 

Ninety days after date, pay to the order of Samuel 
Rogers, seven hundred and fifty dollars value re¬ 

ceived, and charge the same to the account of 

John Dillon. 

To Messrs. Davis &. Tindal, 

Merchants, Philadelphia. 





BUSINESS FORMS. 


329 


No. 2. 

Form of a Bill of Exchange. 

Boston, July 17, 1838 

Exchange for 11,000 francs. 

At sixty days sight of this my first of exchange, 
second and third of the same tenor and date not paid, pay 
to James Parke or order, eleven thousand francs, with or 
without further advice from me. 

Charles Brown. 

Mr. Pierre Davenant, 

Merchant, Paris. 

Note. In order to guard against accident or miscarriage, it is usual 
to draw three copies of a bill of exchange, and send them by different 
conveyances; either of the copies being paid, the others become void. 
The first copy is written like the preceding; in the second, say my 
second of exchange first and third of same tenor and date not paid, ana 
in the third, say my third of exchange , first and second of same tenor 
and dale not paid. 

1. Stephen Lansing wishes for a draft from John Van 
Blarcom of Troy, for $625, on Skinner &, Dustwich of 
Baltimore, payable thirty days after date; how should Van 
Blarcom write it, January 15, 1836 ? 

2. How should Van Blarcom write it so as to be payable 
at sight ? 

Explanation. Say, at sight of this, pay to the order, &c. 

3. How must Horace Harper of Charleston, draw a bill 
of exchange for <£2,000 sterling on Joseph Pettingill of 
London, in favor of Norman Phillips, August 2, 1828, pay¬ 
able thirty days after sight ? 

Explanation. Recollect there are three to be drawn. 

4. How should Sylvanus Brothwick of Baltimore, draw 
a bill of exchange for 10,000 reals on Diego Lana of Ma- 
tanzas, June 20, 1831, to be paid to Manual Morena, or 
bearer, at sight. 

OBSERVATIONS ON DRAFTS AND BILLS OF EXCHANGE. 

A draft or a bill of exchange is nothing but a formal 
order. The distinction between orders, drafts and bills of 


What is done with regard to bills of exchange to guard against acci¬ 
dent or miscarriage ? How is the first copy written ? Second ? Third ? 
What is a draft, or bill of exchange ? 

28* 



330 


BUSINESS FORMS. 


exchange, is this ; orders are used in the transaction of 
business in the neighborhood, dnafts in the transaction of 
business in a distant town or city, and bills of exchange 
in the transaction of business in a foreign nation. More¬ 
over, the presentation, payment, &.C., of drafts, and'espe¬ 
cially bills of exchange, are accompanied with much more 
ceremony than the presentation, payment, &c., of orders. 

Usages with regard to drafts and bills of exchange are 
very much alike. Drafts are sometimes called inland bids 
of exchange , and other bills, foreign bills of exchange. Bills 
of exchange are often bought, sold, and transferred from 
one man to another several times before they become due. 
Banks are in the habit of buying bills. 

The person who makes or draws a bill, is called the 
drawer or seller of the bill. 

The person on whom the bill is drawn, is called the 
drawee ; he is also called the acceptor , if he accepts the 
bill ; that is, agrees to pay it. 

The person to whom the bill is ordered to be paid, is 
called the payee. 

The person who has the bill at any time in his posses¬ 
sion, is called the holder. 

The buyer of the bill who remits it to the drawee, is 
called the remitter. 

When a bill is payable to a certain person or bearer , it 
can be sold, and transferred like an article of merchandise, 
or like a bank note. 

When a bill is payable to a certain person or order , this 
person, who is the payee, can sell and transfer it to whom 
he pleases, by indorsing it, thereby making himself secu¬ 
rity for the payment. 

An indorsement may be blank or special. A blank in- 

What is the distinction between orders, drafts, and bills of exchange ? 
What further difference is there ? 

What is said of the usages with regard to drafts and bills of exchange ? 
What are drafts sometimes called ? Other bills ? What is said of buy¬ 
ing, selling and transferring bills of exchange ? 

What is the person who makes or draws a bill called? 

What is the person on whom the bill is drawn called ? 

What is the person to whom the bill is ordered to be paid oalled ? 

Who is the holder ? The remitter ? 

When a bill is payable to a certain person, or bearer, how can it be 
sold and transferred ? 

When a bill is payable to a certain person, or order, how can it be 
sold and transferred ? 

What may an indorsement be ? 



BUSINESS FORMS. 


331 


dorsement consists of the indorser’s name only, and then 
the bill can be transferred from one person to another by 
simple delivery. A special indorsement is an order from 
the holder directing the money to be paid to a particular 
person, called the indorsee , who must also indorse the bill 
if he sells it. A blank indorsement may always be filled 
by the holder with any person’s name so as to make it 
special. 

Any person may indorse a bill, and every indorser, as 
well as the acceptor is security for the pay ment of the bill. 
An indorsement may be made at any time after the bill is 
drawn, even after the day of payment named in it. 

In reckoning to find when a bill payable in a certain 
number of days after date becomes due, we omit the 
day on which it is dated. When the time is expressed in 
months, calendar months are always understood. 

In the United States and Great Britain three days of 
grace are allowed after the bill is due ; but a bill payable 
at sight, must be paid the day it is presented. 

A bill should be presented for acceptance or payment 
during the usual hours of business ; and a bill should be 
presented for payment on the third day of grace. 

To accept a bill, that is, to agree to pay it, the drawee 
usually writes his name at the bottom, or across the body 
of the bill, with the word accepted. 

When the drawee refuses to accept or to make payment, 
the holder of the bill should give regular and immediate 
notice to all parties to whom he intends to resort for pay¬ 
ment ; for if he does not, they will not be liable to pay. 

With regard to inland bills or drafts, when the drawer 
and drawee live in the same state, the holder has merely 
to send an ordinary notice by letter or otherwise, to the 


What is said of a blank indorsement? What is said of a special 
indorsement? What may always be done to a blank indorsement ? 

May any person indorse a bill ? What is the consequence ? When 
may an indorsement be made ? 

In reckoning to find when a bill payable in a certain number of days 
after date, becomes' due, what do we omit? What if the time is ex¬ 
pressed in months ? 

What is allowed in the United States and Great Britain ? When 
must a bill, payable at sight, be paid ? 

During what hours should a bill be presented for acceptance or pay 
ment ? On what day should a bill be presented for payment? 

How does the drawee usually accept a bill ? 

What if the drawee refuses to accept, or to make payment ? 



332 


BUSINESS FORMS. 


drawer and indorsers, that acceptance or payment, as the 
case may be, is refused, and that he does not intend to 
give credit to the drawee. But with regard to foreign 
bills, or bills from one state to another, a protest is abso¬ 
lutely necessary, and is made as follows. A Public No¬ 
tary appears with the bill, and demands either acceptance 
or payment, as the case may be ; and on being refused, 
draws up an instrument called a protest, expressing that 
acceptance or payment, as the case may be, has been 
demanded and refused, and that the holder of the bill in¬ 
tends to recover any damages he may sustain in conse¬ 
quence. Such an instrument is admitted in all countries 
as a proof of the fact of refusal. 

The protest should be sent as soon as possible to the 
drawer and indorsers, and if it be for non-payment, the bill 
must be sent with the protest. 

If the drawee absconds or cannot be found, protest is to 
be made, and notice given in the same manner as if ac¬ 
ceptance had been refused. 

When acceptance is refused, and the bill is returned by 
protest, an action may be commenced immediately against 
the drawer, though the regular time of payment be not 
arrived. The debt in such case is considered as contract¬ 
ed the moment the bill is drawn. 

The preceding observations contain some of the princi¬ 
pal rules which govern merchants in the use of bills ; the 
other laws and customs on this subject are very numerous, 
and must be learned by practice. 


How does he give notice with regard to inland bills or drafts, when 
the drawer and drawee live in the same state ? What is necessary with 
regard to foreign bills, or bills from one state to another ? How is a pro¬ 
test made r What is this instrument admitted as a proof of? 

What should be done with the protest and bill ? 

What if the drawee absconds, or cannot be found ? 

When may an action be commenced against the drawer ? When is 
the debt in such a case considered as contracted ? 



NOTES. 


333 


NOTES. 

Note 1. 

Signs. 

Certain algebraic signs are employed in many arithmetics, for the pur¬ 
pose, it is said, of abridging common language. They are never used ir 
the transaction of business, but as they are occasionally seen in books 
the following description of them should be studied after Division. 
The scholar should then be occasionally required to explain by these 
signs how he performs an example. 

= Two horizontal lines are the sign of equality. They show that 
numbers with this sign between them are equal; thus, 5 added to 3 = 8. 
They may be read, is equal to ; thus, 5 added to 3 = 8, may be read, 5 
added to 3 is equal to 8. 

4- An upright cross is the sign of addition. It shows that two num¬ 
bers with this sign between them are added together; thus, 5 -J- 3 = 8. 
It may be read, added to ; thus, 5 -f- 3 = 8, may be read, 5 added to 3 is 
equal to 8. 

This sign is sometimes put at the right of decimals, to show that the 
entire number is not written, the figures neglected being of small 
value; thus, the value of J being in decimals .3333, &c., we can write 
333-f—, instead of writing .333 about. 

— A horizontal line is the sign of subtraction. It shows that the 
number after it is to be subtracted from the number before it; thus, 
5 — 3 = 2. It may be read, less by; thus, 5 — 3 = 2 may be read, 
5 less by 3 is equal to 2. 

This sign is sometimes put at the right of decimals, to show that a 
number a little too large is written, the last figure being increased by 1, 
because the first one neglected is 5 or more ; thus, the value of § in 
decimals being .6(366, &c., we can write .667—, instead of writing .667 
nearly. 

X An inclined cross is the sign of multiplication ; it shows that two 
numbers with this sign between them, are multiplied together; thus, 
3 X 5 = 15. It may be read, multiplied by ; thus, 3 X 5 = 15, may be 
read, 3 multiplied by 5 is equal to 15. 

-f- A horizontal line with a dot above it and another below it, is the 


For what purpose are certain algebraic signs employed in many arith¬ 
metics? Are they ever used in the transaction of business? 

What are two horizontal lines ? What do they show? Give an ex¬ 
ample. How may they be read ? Give an example. 

What is the sign of addition ? What does it show ? Give an exam¬ 
ple. How may it be read? Give an example. 

Where is this sign sometimes put, and for what purpose ? Give an 
example. 

What is the sign of subtraction? What does it show ? Give an ex¬ 
ample. How may it be read ? Give an example. 

Where is this sign sometimes put, and for what purpose ? Give an 
example. 

What is the sign of multiplication ? What does it show ? Give an 
example ? How may it be read ? Give an example. 

What is the sign of division ? 



334 


NOTES. 


sign of division; it shows that the number before it is divided by the 
number after it; thus, 15 = by 3 = 5. It may be read, divided by; 
thus, 15 = 3 = 5 may be read, 15 divided by 3 is equal to 5. 

EXERCISES. 

How do you read 4 -f 9 = 13? 16 — 7 = 9 ? 11 X 5 = 55 ? 24-4-8 
= 3? 18 + 25=43? 20 — 5 = 15? 15 x 15 = 225? 320 = 16 

= 20? 6+2 = 15 —7? 39 —4 = 7 X 5? 4+5 = 99=11? 

How do you write by means of the preceding signs, 6 and 7 are 13, 
that is, 6 added to 7 is equal to 13 ? 11 from 23 leaves 12 ? 25 times 33 
are 825 ? 7 is in 385, 55 times ? 20 and 45 are 65 ? 73 from 137 leaves 
64 ? 12 times 12 are 144 ? 16 is in 480, 30 times ? 9 added to 5 is 
equal to 20 less by 6 ? 27 added to 5 is equal to 4 multiplied by 8 ? 11 
less by 2 is equal to 108 divided by 12 ? 

Many numbers are sometimes connected together by these signs; 
thus, 5 + 8 — 4 X 8 = 6 = 9 x4-t-12x 4. These numbers may be 
read, 5 added to 8, the result less by 4, this result multiplied by 8, and 
this result divided by 6, is equal to 9 multiplied by 4, the result 
divided by 12, and this result multiplied by 4. 

Though these signs are of no advantage in arithmetic they are very 
useful in algebra, indeed algebra is a language of signs, but then they 
are usually employed for different purposes from what they are in arith¬ 
metic. 


Note 2. 

Repeating Decimals, often called Circulating Decimals. 

Decimals consisting of figures continually repeated, are called repeat¬ 
ing decimals or repetends ; as .333, &c., .2121, &c. 

When only one figure is repeated the fraction is called a single repe¬ 
tend, as .333, &c. 

When two or more figures are repeated the fraction is called a com¬ 
pound repetend ; as .2121, &c. 

When other decimals come before the repeating decimals the fraction 
is called a mixed repetend ; thus, .1666, &c. is a mixed single repetend, 
and .2304304, &c. is a mixed compound repetend. 

Mathematicians sometimes write but one figure in a single repetend, 
and place a dot over it; thus, .333, &c. is written .3. In a compound 
repetend they write the repeating figures but once, and place a dot over 
the first and last; thus, .2121, &c. is written .2i, and .2304304, is writ¬ 
ten .2304. 

What does it show ? Give an example. How may it be read ? Give 
an example. 

Are these signs of any advantage in arithmetic ? What is said of 
them in algebra ? 

What are called repeating decimals or repetends ? Give some exam¬ 
ples. 

What is called a single repetend ? Give an example. 

What is called a compound repetend ? Give an example. 

What is called a mixed repetend? Give examples of a mixed single 
repetend, and of a mixed compound repetend. 

How do mathematicians sometimes write a single repetend? Give an 
example. How do they write a compound repetend ? Give some ex 
amples. 




.NOTES. 


335 

For all practical purposes, repeating decimals can be employed just 
like any other decimals. But as some of them must always be omitted 
in calculations, the results will not be perfectly correct. 

Let us see if we can change them to common fractions with perfect 
accuracy. r 

Changing § to decimals we get . 1111 , &c. or . 1 . 

So . 1111 , &c. or .1 is | in common fractions. Also .2, or 2 times 1 
is 1 in common fractions, .3, or 3 times .i is f, .4 is f, .5 is §, &c. 

Changing § § to decimals we get .0101, &c. or 6 i. 

, *0101, &c. or .01 is 55 in common fractions. Also .02, or 2 times 

.01 is 55 in common fractions, .03, or 3 times .01 is 5 3 5 , .04 is § 4 5 , .15 is |f, 
.76 is 55 , &c. 

Changing 5 | 5 to decimals we get .001001, &c. or .00i. 

So .001001, &c. or .00i is 555 in common fractions. Also .002, or 2 
times .001 is §|§ in common fractions, .003, or 3 times . 00 i is 555 , .004 

is 555 , .026 is 55 ® 5 , .235 is 555 , &c. 

Therefore, to change a repeating decimal to a common fraction, 

Make the repetcnd a numerator , and write as many 9s for a denomina¬ 
tor as there are repeating figures. 

EXAMPLES. 

What common fraction is equal to .3, .5, . 8 , .36, .49, .137, .403, .7. .12, 
.4, .737, .8534, .82, . 6 , .i23. 

We can now change a mixed repetend to a'common fraction with 
ease. 

For example, to change .16 to a common fraction, we proceed thus; 
Value of .1. Value of .06. Explanation. .16 is composed of .1 
15 5 S and .06; the value of .1 in common 

Changing to a common de- fractions is j 0 , and the value of .06 is 
nominator^ and adding, we get, evidently 5 % since .06 is one tenth of 
or g Ans. (j w hose value is 5 . Adding to 53 , 

we get Icq or J. 

After changing repeating decimals to common fractions, we can add, 
subtract, multiply, and divide in them without any necessary errors. 

EXAMPLES. 

1 . Add .5, .83, .625, and . 6 . Ans. 2.625. 

2. Subtract .735 from 2.3. Ans. lfff or 1.597. 

3. Multiply .03 by .2. Ans. g®i- 

4. Divide .4 by .i23. Ans. 

For all practical purposes, how can repeating decimals be employed? 
Will the results be perfectly correct? Why? 

Explain how we can change repeating decimals to common fractions 
with perfect accuracy. 

How do we change a repeating decimal to a common fraction ?. 
Explain how we proceed to change a mixed repetend, say 16, to a 
common fraction ? 

After changing repeating decimals to common fractions what can we 





336 


NOTES. 


Note 3. 


Duodecimals. 

Arithmetics usually contain a rule called Duodecimals. It shows 
how to multiply feet and inches together without changing them to the 
same denomination. The rule is difficult and useless, but as many per¬ 
sons employ this method in measuring wood, the following description 
of it should be studied. 


1. What is the surface of the end of a load of wood 3 ft. 10 in. wide, 
and 4 ft. 8 in. high ? 


OPERATION. 

ft. in. 

4 8 

3 10 


3 10 8 

14 0 


17 10 8 

Ans. 17 sq. ft. and 


Explanation. Multiplying 8 in , or ig of 
a foot, by 10 in., or i§, we get or j 6 g and 
jfs; we put down the if?, and multiplying 
4 ft. by {%, get fg, to which we add the T g, and 
obtain or 3 ft. i|. Now multiplying the 
^ by 3 ft., we get ff, or 2 ft. 5 °g; we put 
down the 0 under the 12ths, and add the 2 
ft. to the 12 ft. found by multiplying the 4 ft. 
by 3 ft. 


2. What quantity of wood is there in the preceding load, it being 
7 ft. 1 in. long? 


OPERATION. 

sq. ft. 

17 10 8 surface of end. 

7 1 


1 5 10 8 

125 2 8 

126 8 6 8 

Ans. 126 cubic ft. fg, and 

In Duodecimals, therefore, 


Explanation. We proceed as in 
the preceding example, carrying 
the 12s in each column one place 
to the left. 


Proceed as in decimal numbers , but carry 1 for every 12 instead of 1 
for every 10. 

Most arithmetics require the scholar to multiply in Duodecimals first 
by the feet, and then to put the product by the inches beneath the first 
product one place to the right. For instance; 


What does the rule called Duodecimals show ? Is the rule easy and 
useful ? 

Explain how example 1 is performed. 

Explain how example 2 is performed. 

How do we proceed in Duodecimals ? 

How do most arithmetics require the scholar to multiply in Duodeci¬ 
mals ? 









NOTES. 


337 


Example 1 is performed thus, 

and example 2 thus; 

4 8 

17 10 8 

3 10 

7 1 

14 0 

125 2 8 

3 10 8 

1 5 10 8 

17 10 8 

126 8 6 8 

This course evidently gives the 

same result as before. It produces 

no benefit however, but renders the 

process more obscure. 

3. What is the value of the preceding load of wood at $£ a ft. ? 

OPERATION. ^ 

12)8 1728ths. 

16)126.7129(7.92 ft. of wood. 

112 - 

12 )6.666, &c. 144ths. 

- $3.96 value. Ans. 

147 

12)8.555, &c. 12ths. 

144 


126.7129 cubic ft. 31 

32 

The long and difficult operations in examples 1, 2, and 3, are easily 
performed by decimals, according to lesson 173. 

4.7 height. 

3.8 width. 16)126.806(7.92 ft. of wood. 

- 112 - 

376 - $3.96 value. Ans. 

141 148 

- 144 

17.86 sq. ft., surface of end. - 

7.1 length. 40 

■ 32 

1786 — 

12502 


126.806 cubic ft. in load. 

In many cases it is necessary to take the dimensions within less than 
an inch, and then the operation by duodecimals becomes so tedious that 
decimals must be employed. 


EXAMPLE. 

4. What is the value of a polished marble block 5.45 ft. or 5 ft. 5§ in. 

long, 3.64 ft., or 3 ft. 7g in. wide, and 2.99 ft. or 2 ft. 11J in. high, at $4 
a cubic foot ? Ans. $237.26. 

Perform the following examples by duodecimals. 

5. Example 6, lesson 107. Ans. 26.58, about. 

Ans. 405. 

Ans. 7 C. 13.5 cubic ft. 

Ans. .8125. 

Ans. $16.05. 


6 . Example 7, lesson 107. 

7. Example 8, lesson 107. 

8 . Example 9, lesson 107. 

9. Example 7, lesson 173. 


What is said of this course ? 

By what are the long and difficult operations in examples 1,2 and 3 
easily performed ? 

In many cases how is it necessary to take the dimensions ? What is 
the consequence P 

29 











338 


NOTES. 


Note 4. 

Proportion. 

In the Rule of Three the examples which are performed so easily by 
common sense, are often solved in a difficult manner, by means of alge¬ 
braic proportion. As some persons employ this method, the following 
description of it should be studied, and used in solving examples in the 
Rule of Three, and in the other rules to which it may be applied. 

1. If 2 pears cost 6 cents, what will 4 pears cost ? A ns. 12 cents. 

In this example we see 2 pears bear the same proportion to 4 pears 
as 6 cents to 12 cents, and that a similar proportion must exist in like 
cases. For the second lot of pears being 2 times the first, the price 
must be 2 times the price of the first; likewise, if the second lot had 
been 3 times the first, the price would have been 3 times the price of the 
first; if the second lot had been 2| times the first, the price would have 
been 2\ times the price of the first; if the second lot had been g of the 
first, the price would have been J of the price of the first, &c. 

To show that the two lots of pears bear the same proportion to each 
other as their prices, mathematicians write the numbers thus ; 

pears, pears. cents, cents. 

2 : 4 : : 6 : 12 

and read them, so written, thus; 2 pears are to 4 pears as 6 cents are to 
12 cents. 

Any four numbers in which the first bears the same proportion to the 
second as the third does to the fourth, when written like the preceding, 
form an algebraic proportion, commonly called a proportion. 

We see (hat the ratio of 2 to 4 is |, or |, and is the same as the ratio 
of 6 to 12, which is T 6 _, or |. So a proportion is formed of two equal 
ratios. 

The four numbers constituting a proportion, as 2, 4, 6, and 12 in the 
preceding one, are called the terms. 

The two outside terms, as 2 and 12, are called the extremes . 

The two middle terms, as 4 and 6, are called the means. 

The two first terms of the ratios, as 2 and 6, are called the antecedents. 

The two last terms of the ratios, as 4 and 12, are called the conse¬ 
quents. 

In the preceding proportion the ratio of the first term to the second is 


What is Said of performing the examples in the Rule of Three by 
Common sense, and by algebraic proportion ? 

In example 1, what proportion do 2 pears bear to 4 pears? What 
must exist in like cases ? Why ? 

How do mathematicians show that the two lots of pears bear the same 
proportion to each other as their prices ? How do they read them so 
written ? 

What form an algebraic proportion ? 

What is the ratio of 2 to 4, and what is it the same as? What then 
is a proportion formed of? 

What are called the terms? The extremes? The means? The 
antecedents ? The coriseqnents ? 

In the preceding proportion what is the ratio of the first term to the 
secOijd ? Of the third term to the fourth ? 



NOTES. 


339 


|, and the ratio of the third term to the fourth, is Changing these 

fractions to a common denominator, we get, 

24 and 24 

48 48 

In changing to a common denominator, the numerator of the first 
fraction is formed by multiplying 2 and 12, or the two extreme^ of the 
proportion ; and the numerator of the second fraction is formed by mul¬ 
tiplying 4 and 6, or the two means. Now the two ratios forming a pro¬ 
portion always being equal, the two fractions, when changed to a 
common denominator, must always have equal numerators. 

Therefore in a proportion, 

The product of the tico extremes is equal to the product of the two means. 
So if we divide the product of the two means by one extreme , the quotient 
will be the other. 

In order, then, to perform example 1 by algebraic proportion, we write 
the first three terms thus ; 

2 : 4 : : 6 

and to get the fourth term, or the answer, we divide the product of 4 
and 6 by 2, thus; 

6 

J 

2)24 

12 cents, fourth term, or A ns. 

2. If 3 men can build a stone wall in 4 days, how long will it take 6 
men to build it ? A ns. 2 days. 

In example 2 we see that 

• men. men. days. days. 

6 : 3 : : 4 : 2 

To perform example 2 by algebraic proportion, we write the first three 
terms thus ; 

6 : 3 : : 4 

and to get the fourth term, or the answer, we divide the product of 3 
and 4 by 6, thus; 

4 

3 

6)12 

2 days, fourth term, or Ans. 

The proportion under example 2 is evidently correct, and a similar 
one must exist in like cases. For the second lot of men being 2 times 
the first, can perform the labor in \ of the time ; likewise if the second 


In changing these fractions to a common denominator, how is the 
numerator of the first fraction formed ? How is the numerator of 
the second fraction formed ? Why must the two fractions, when 
changed to a common denominator, always have equal numerators? 

In a proportion, what is the product of the two extremes equal to? 
How do we get one of the extremes ? 

How then do we perform example 1 by algebraic proportion ? 

In example 2 what do we see ? 

How do we perform example 2 by algebraic proportion ? 

Why is the proportion under example 2 correct, and why must a sim¬ 
ilar one exist in like cases ? 



340 


NOTES. 


lot had been 3 times the first, they could have performed the labor in $ 
of the time; if the second lot had been 2 § times the first, or 8 , they 
could have performed the labor in § of the time; if the second lot had 
been J of the first, they could have performed the labor in 3 times the 
time. 

In example 1, the more pears the more they cost, but in example 2, 
the more men the fewer days they will be in doing the work. The 
principle in the first proportion is inverted in the second; so the first is 
called a direct proportion , and the second an inverse proportion. 

By examining the proportions under examples 1 and 2, we see that the 
third terms are of the same nature or kind as the answer. Then if the 
answer must be larger than the third term, the second term is larger 
than the first, but if the answer must be smaller than the third term, 
the second term is smaller than the first. It is generally so in other 
cases. 

From what precedes we derive the following rule for performing an 
example in the Rule of Three, by algebraic proportion. 

Make that number ichich is of the same kind as the ansicer sought the 
third term of the proportion. Then if the answer must be larger than the 
third term , make the larger of the other tico numbers the second term, and 
the smaller the first term , but if the answer must be smaller than the third 
term , make the smaller of the other two numbers the second term , and the 
larger the first term. Afterwards, to get the answer, divide the product 
of the second and third terms by the first. 

Algebraic proportion can also be employed in solving many of the 
examples in Percentage, Commission, Stocks, Bankruptcy, Loss and 
Gain, Duties, Interest, Rule of Three Compound, Barter, Fellowship, 
Insurance, and Simple Machines, as follows. 

3 Percentage, lesson 121, example 1. If the man must invest $4 
per $100 in books, what must he invest out of $225 1 


100 : 225 : • 4 
4 

1100)9100 $9, Ans. 

4. Interest, lesson 128, example 1. If the interest be $6 per $100, 
r one year, what will it be on $100.33? 

100 : 100.33 : : 6 6.0198 6.0 2 

6 2 ^ yrs. 15 

1|00)6|0.198 12.0396 30T0 

Add 25 cents 6 0 2 


$12-29 Ans. 3 65)9 0.3 0(25 cents. 
730 
1730 
1825 


What is the difference between the proportion in example 1 and the 
proportion in example 2 ? What is the first called ? The second ? 

By examining the proportions under example 1 and 2, what do we see ? 
From what precedes; what rule do we derive for performing an ex¬ 
ample in the Rule of Three by algebraic proportion ? 

For what can algebraic proportion also be employed ? 









NOTES. 


341 


5. Discount, example 3. If the interest of $100 for 1 year be $7, 
what will it be for 2 years ? Ans. $14. 

100 

Now if $114 payable in 2 years, is worth only 
$100, what is $500, payable in the same time, worth ? 

114 : 500 : : 100 

100 $ 

114)50000(438.00 Ans. 

In the Rule of Three Compound two statements are made. 

6. Rule of Three Compound, lesson 140, example 1. If it takes 
2 years for the interest of $150 to amount to $18, how long will it take 
for the interest of $675 to amount to the same sum ? 

675 : 150 : : 2 
2 

300 

- or | of a year. 

dividing by 675 675 

Now if it takes $675 f of a year to amount to $18, how long will it 
take it to amount to $ 162 ? 

18 . : 162 : : f 

4 

9)648 

18) 72(4 years. Ans. 

72 


7. Barter, lesson 152, example 9. If 36 cents are demanded for 
what is worth 30, how much should be asked for what is worth 15 ? 


30 : 15 : : 36 

15 

• 180 
36 

3)0)5410 

18 cents, Ans. 

8 . Simple Fellowship, example 1. 

1000 

1250 


Now if $2250 gains $945, what will $1000 gain ? 
2250 : 1000 : : 945 

1000 $ 

2250)945000(420 A.’s dividend, &c. 
9. Compound Fellowship, example 1. 

400 450 

2400 _6 _8 

3600 2400 3600 

Now if 6000 gains $120, what will 2400 gain ? 


What is done in the Rule of Three Compound ? 

29 # 





NOTES. 


342 


6000 : 120 : : 2400 

120 

48 

24 

61000) 288)00 0 

$48 A.’s share, Ans. 

10. The Lever, lesson 196, example 2. 

720 : 50 : : 12 
50 

720)600(0 feet 
12 
12 
6 

7200(10 inches, Ans. 

In many examples, however, it is by the rule difficult or impossible to 
state the question. Such examples cannot be performed by the rule , but 
must be solved by common sense. 

The following are examples of this kind. 

11. Four boys, A, B, C, and D, have a number of cents; A has 1 
cent, B 2 cents, and C 3 cents ; now D’s money bears the same propor¬ 
tion to C’s as B’s does to A’s ; how many cents has D ? 

12. A bin containing 60 bu. of wheat diminished by drying to 58.5 
bu.; how much then should a bin containing 400 bu. diminish from the 
same cause ? 

13. There are four numbers, the second bearing the same proportion 
to the first, as the fourth does to the third ; the first number is 22, the 
second is 2, and the fourth 11; what is the third ? 


In many examples, however, what is it difficult or impossible to do 
by the rule ? What is said of such examples ? 


* 

THE END. 































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